1. Discontinuous Galerkin methods for approximating the solutions to problems in nonlinear elasticity [2008]
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3. Dislocation dynamics of face-centered cubic metals and alloys [electronic resource] [2016] Online
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- 1 online resource.
Face-centered cubic (FCC) metals and alloys—such as aluminum, copper, and austenitic stainless steel—are ubiquitous in the automotive, aerospace, and oil and gas industries. To further our understanding of the nature of plastic deformation in these materials, we have utilized dislocation dynamics (DD) simulations. We begin with a study of time integration in dislocation dynamics, examining the time-step-limiting aspects of DD, and developing a new subcycling-based time integrator that improves efficiency 100-fold. We then utilize this time integrator to study the basics of plasticity in pure, single crystalline copper. DD simulations were run over a wide range of strain rates and initial dislocation densities, examining how the yield strength and hardening rate vary, and making comparisons against the available experimental data. A detailed study on the contribution of binary dislocation junctions to hardening is then presented, showing that these junctions are an essential ingredient for hardening to occur. We then go on to study the most common strengthening mechanisms employed in FCC metals: solid solution strengthening and precipitation strengthening. After deriving the suitable chemical potential for the solute atoms comprising a solid solution, the influence of an "atmosphere" of solutes surrounding a dislocation on the stress field and line tension of the dislocation are examined, showing that these effects are generally small. The drag force exerted on dislocations by their atmospheres is then studied. We find these drag forces are often larger than those due to lattice friction, and can influence plasticity in FCC materials significantly. To study precipitation strengthening, a new algorithm for simulating dislocation-precipitate interactions is developed, which allows for ellipsoidal inclusions with arbitrary aspect ratio and arbitrary misfit. The new formulation is used to study Orowan looping in overaged aluminum-copper alloys, which have plate-like precipitates. The results and methods presented here constitute a broad set of advancements towards a more sound understanding of plastic deformation in FCC metals and alloys.
Face-centered cubic (FCC) metals and alloys—such as aluminum, copper, and austenitic stainless steel—are ubiquitous in the automotive, aerospace, and oil and gas industries. To further our understanding of the nature of plastic deformation in these materials, we have utilized dislocation dynamics (DD) simulations. We begin with a study of time integration in dislocation dynamics, examining the time-step-limiting aspects of DD, and developing a new subcycling-based time integrator that improves efficiency 100-fold. We then utilize this time integrator to study the basics of plasticity in pure, single crystalline copper. DD simulations were run over a wide range of strain rates and initial dislocation densities, examining how the yield strength and hardening rate vary, and making comparisons against the available experimental data. A detailed study on the contribution of binary dislocation junctions to hardening is then presented, showing that these junctions are an essential ingredient for hardening to occur. We then go on to study the most common strengthening mechanisms employed in FCC metals: solid solution strengthening and precipitation strengthening. After deriving the suitable chemical potential for the solute atoms comprising a solid solution, the influence of an "atmosphere" of solutes surrounding a dislocation on the stress field and line tension of the dislocation are examined, showing that these effects are generally small. The drag force exerted on dislocations by their atmospheres is then studied. We find these drag forces are often larger than those due to lattice friction, and can influence plasticity in FCC materials significantly. To study precipitation strengthening, a new algorithm for simulating dislocation-precipitate interactions is developed, which allows for ellipsoidal inclusions with arbitrary aspect ratio and arbitrary misfit. The new formulation is used to study Orowan looping in overaged aluminum-copper alloys, which have plate-like precipitates. The results and methods presented here constitute a broad set of advancements towards a more sound understanding of plastic deformation in FCC metals and alloys.
4. Mapped finite element methods with applications to the simulation of brittle fracture propagation [electronic resource] [2016] Online
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- 1 online resource.
The thesis presents the development, the mathematical analysis, as well as applications of a computational framework for the simulation of curvilinear crack propagation. At the core of the computational framework lies a novel finite element method, named the Mapped Finite Element Method (MFEM), for the optimal convergence of singular solutions. The main challenges in solving numerically the mathematical description of a propagating fracture can be identified in: the continuously evolving (cracked) domain, the singular nature of the elasticity fields, and the computation of the stress intensity factors for the prediction of crack growth. Current state-of-the-art methods are plagued by low order of accuracy, high computational cost, and complex data structures. The work herein addresses the aforementioned challenges by developing a computationally efficient, rapidly convergent, and non-intrusive algorithm consisting of three key ingredients: Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals. First, Universal Meshes are introduced as a computationally efficient and robust meshing algorithm for the generation of conforming subdivisions of the evolving domain. Second, Mapped Finite Element Methods (MFEM) will be developed for the solution of the singular elasticity fields. The methods are shown to converge with optimal order for the same computational cost, preserving well conditioning and sparsity properties, and with no alteration to the data structure of standard Lagrange finite element methods (known to converge sub-optimally for this class of problems). The optimality of convergence is supported by mathematical analysis and applications of MFEM are showcased beyond brittle fracture (e.g. the resolution of boundary layers in flows around moving obstacle, real-space Kohn-Sham density functional theory calculations, etc). Third, the thesis presents the construct of a family of linear and affine functionals, named Interaction Integral functionals, for the rapidly convergent computation of the stress intensity factors (SIFs) for curvilinear fractures. The distinct feature of the Interaction Integral functionals is their ability to double the rate of convergence of the energy norm of the solution in the evaluation of the SIFs. Sketches of the mathematical analysis are provided to support the observed rapid rates of convergence. The propagation algorithm that combines the developed tools (Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals) is presented and shown to be consistent (in the sense of being able to replicate observed experimental results) and predictive (in the sense of yielding crack paths that converge to a unique solution with refinement of the discretization). Lastly, the capabilities of the developed algorithm are exploited to study the formation of wavy crack patterns when brittle heat conductors are rapidly cooled.
The thesis presents the development, the mathematical analysis, as well as applications of a computational framework for the simulation of curvilinear crack propagation. At the core of the computational framework lies a novel finite element method, named the Mapped Finite Element Method (MFEM), for the optimal convergence of singular solutions. The main challenges in solving numerically the mathematical description of a propagating fracture can be identified in: the continuously evolving (cracked) domain, the singular nature of the elasticity fields, and the computation of the stress intensity factors for the prediction of crack growth. Current state-of-the-art methods are plagued by low order of accuracy, high computational cost, and complex data structures. The work herein addresses the aforementioned challenges by developing a computationally efficient, rapidly convergent, and non-intrusive algorithm consisting of three key ingredients: Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals. First, Universal Meshes are introduced as a computationally efficient and robust meshing algorithm for the generation of conforming subdivisions of the evolving domain. Second, Mapped Finite Element Methods (MFEM) will be developed for the solution of the singular elasticity fields. The methods are shown to converge with optimal order for the same computational cost, preserving well conditioning and sparsity properties, and with no alteration to the data structure of standard Lagrange finite element methods (known to converge sub-optimally for this class of problems). The optimality of convergence is supported by mathematical analysis and applications of MFEM are showcased beyond brittle fracture (e.g. the resolution of boundary layers in flows around moving obstacle, real-space Kohn-Sham density functional theory calculations, etc). Third, the thesis presents the construct of a family of linear and affine functionals, named Interaction Integral functionals, for the rapidly convergent computation of the stress intensity factors (SIFs) for curvilinear fractures. The distinct feature of the Interaction Integral functionals is their ability to double the rate of convergence of the energy norm of the solution in the evaluation of the SIFs. Sketches of the mathematical analysis are provided to support the observed rapid rates of convergence. The propagation algorithm that combines the developed tools (Universal Meshes, Mapped Finite Element Methods, and Interaction Integrals) is presented and shown to be consistent (in the sense of being able to replicate observed experimental results) and predictive (in the sense of yielding crack paths that converge to a unique solution with refinement of the discretization). Lastly, the capabilities of the developed algorithm are exploited to study the formation of wavy crack patterns when brittle heat conductors are rapidly cooled.
5. Design and analysis of numerical methods for free- and moving-boundary problems [electronic resource] [2015] Online
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- 1 online resource.
This thesis presents the design and analysis of numerical methods for free- and moving-boundary problems: partial differential equations posed on domains that change with time. Two principal developments are presented. First, a novel framework is introduced for solving free- and moving-boundary problems with a high order of accuracy. This framework has the distinct advantage that it can handle large domain deformations easily (a common difficulty faced by conventional deforming-mesh methods) while representing the geometry of the moving domain exactly (an infeasible task for conventional fixed-mesh methods). This is accomplished using a universal mesh: a background mesh that contains the moving domain and conforms to its geometry at all times by perturbing a small number of nodes in a neighborhood of the moving boundary. The resulting framework admits, in a general fashion, the construction of methods that are of arbitrarily high order of accuracy in space and time when the boundary evolution is prescribed. Numerical examples involving phase-change problems, fluid flow around moving obstacles, and free-surface flows are presented to illustrate the technique. Second, a unified analytical framework is developed for establishing the convergence properties of a wide class of numerical methods for moving-boundary problems. This class includes, as special cases, the technique described above as well as conventional deforming-mesh methods (commonly known as arbitrary Lagrangian-Eulerian, or ALE, schemes). An instrumental tool developed in this analysis is an abstract estimate, which applies to rather general mesh motions, for the error incurred by finite element discretizations of parabolic moving-boundary problems. Specializing the abstract estimate to particular choices of the mesh motion strategy and finite element space leads to error estimates in terms of the mesh spacing for various semidiscrete schemes. We illustrate this by deriving error estimates for ALE schemes under mild assumptions on the nature of the mesh deformation and the regularity of the exact solution and the moving domain, and we do the same for universal meshes.
This thesis presents the design and analysis of numerical methods for free- and moving-boundary problems: partial differential equations posed on domains that change with time. Two principal developments are presented. First, a novel framework is introduced for solving free- and moving-boundary problems with a high order of accuracy. This framework has the distinct advantage that it can handle large domain deformations easily (a common difficulty faced by conventional deforming-mesh methods) while representing the geometry of the moving domain exactly (an infeasible task for conventional fixed-mesh methods). This is accomplished using a universal mesh: a background mesh that contains the moving domain and conforms to its geometry at all times by perturbing a small number of nodes in a neighborhood of the moving boundary. The resulting framework admits, in a general fashion, the construction of methods that are of arbitrarily high order of accuracy in space and time when the boundary evolution is prescribed. Numerical examples involving phase-change problems, fluid flow around moving obstacles, and free-surface flows are presented to illustrate the technique. Second, a unified analytical framework is developed for establishing the convergence properties of a wide class of numerical methods for moving-boundary problems. This class includes, as special cases, the technique described above as well as conventional deforming-mesh methods (commonly known as arbitrary Lagrangian-Eulerian, or ALE, schemes). An instrumental tool developed in this analysis is an abstract estimate, which applies to rather general mesh motions, for the error incurred by finite element discretizations of parabolic moving-boundary problems. Specializing the abstract estimate to particular choices of the mesh motion strategy and finite element space leads to error estimates in terms of the mesh spacing for various semidiscrete schemes. We illustrate this by deriving error estimates for ALE schemes under mild assumptions on the nature of the mesh deformation and the regularity of the exact solution and the moving domain, and we do the same for universal meshes.
- Book
- 1 online resource.
Three-dimensional realistic simulations are often unsteady and involve moving geometries. Problems of this type include design-shape optimization, crack propagation, hydraulic fracturing, solid dynamics for large deformations, phase transformations, and fluid-structure interaction, among others. Finite element methods for such problems fall in two different classes: 1) deforming mesh methods, and 2) immersed boundary methods. Deforming mesh methods require remeshing or updating the mesh for finite element calculations every few time steps/instances. Immersed boundary methods approximate the changing domain through a fixed background mesh. In this thesis we present a mesh generation algorithm that retains the feature of generating conforming meshes for changing domains, and is computationally efficient like the immersed boundary methods. The method we propose can be used with any off-the-shelf solver for moving domain three dimensional simulations. We introduce a method to mesh the boundary of a smooth, open domain in three dimensions immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to . Two types of surface meshes follow: (a) a mesh that exactly meshes the surface, and (b) meshes that approximate the surface to any order, by interpolating the map over the selected faces; i.e., an isoparametric approximation to the surface. The map we use to deform the faces is the closest point projection onto the surface. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedra should: (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We extend the method to mesh the boundary, to generate a conforming mesh of tetrahedra for the open domain and bounded domain in three dimensions. Similarly to the case of surface meshes, we generate tetrahedral meshes that approximate the domain to any order, by interpolating the map over the selected tetrahedra. Through the perturbation of selected vertices in the background mesh, we construct a mapping that ensures tetrahedra with positive measure in the approximating tetrahedralization. An approximating tetrahedral mesh for a given domain is constructed by : a) selecting a set of vertices in a neighborhood of the surface and b) perturbing them. We need the tetrahedra in the neighborhood of the surface to satisfy the conditions necessary for quality-ensured surface meshes. The perturbation is described in two different ways: 1) explicit mapping, and 2) quality optimization based perturbations. We present an efficient implementation of the algorithm to generate approximate tetrahedralization that takes advantage of the conjecture that the set of faces used for the surface mesh is a connected set. We present several numerical experiments that are consistent with this conjecture. These experiments suggest that the complexity of the presented algorithm is linearly proportional to the number of triangles in the surface mesh of the surface. We showcase the usage of the algorithm with various examples of moving domain problems. We further extend the notion of universal meshes to generate a conforming triangulation for a given smooth curve from a generalized anisotropic background mesh. In particular method allows for triangles with obtuse angles in the background mesh. The novelty of the approach lies in the way the set of faces are selected to discretize the curve, and in the projection map constructed for the selected faces that discretize the given curve. We discuss two particular cases where the proposed method fails to generate conforming meshes. We generate these conforming meshes without altering the connectivity of the background mesh. Furthermore, we present preliminary results of using the same method for continuous curves, and anisotropic tetrahedral meshes for three-dimensional geometries. The universal meshes algorithm presented here is advantageous for a few important reasons. The method does not require that any global problem be solved; the background mesh is only generated once; and the connectivity of the background mesh is left unaltered. Hence, the computational complexity of mesh generation is observed to be proportional to the number of triangles on the surface. The examples and discussion presented here indicate that universal meshes can be a useful tool in simulating realistic engineering problems.
Three-dimensional realistic simulations are often unsteady and involve moving geometries. Problems of this type include design-shape optimization, crack propagation, hydraulic fracturing, solid dynamics for large deformations, phase transformations, and fluid-structure interaction, among others. Finite element methods for such problems fall in two different classes: 1) deforming mesh methods, and 2) immersed boundary methods. Deforming mesh methods require remeshing or updating the mesh for finite element calculations every few time steps/instances. Immersed boundary methods approximate the changing domain through a fixed background mesh. In this thesis we present a mesh generation algorithm that retains the feature of generating conforming meshes for changing domains, and is computationally efficient like the immersed boundary methods. The method we propose can be used with any off-the-shelf solver for moving domain three dimensional simulations. We introduce a method to mesh the boundary of a smooth, open domain in three dimensions immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to . Two types of surface meshes follow: (a) a mesh that exactly meshes the surface, and (b) meshes that approximate the surface to any order, by interpolating the map over the selected faces; i.e., an isoparametric approximation to the surface. The map we use to deform the faces is the closest point projection onto the surface. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedra should: (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We extend the method to mesh the boundary, to generate a conforming mesh of tetrahedra for the open domain and bounded domain in three dimensions. Similarly to the case of surface meshes, we generate tetrahedral meshes that approximate the domain to any order, by interpolating the map over the selected tetrahedra. Through the perturbation of selected vertices in the background mesh, we construct a mapping that ensures tetrahedra with positive measure in the approximating tetrahedralization. An approximating tetrahedral mesh for a given domain is constructed by : a) selecting a set of vertices in a neighborhood of the surface and b) perturbing them. We need the tetrahedra in the neighborhood of the surface to satisfy the conditions necessary for quality-ensured surface meshes. The perturbation is described in two different ways: 1) explicit mapping, and 2) quality optimization based perturbations. We present an efficient implementation of the algorithm to generate approximate tetrahedralization that takes advantage of the conjecture that the set of faces used for the surface mesh is a connected set. We present several numerical experiments that are consistent with this conjecture. These experiments suggest that the complexity of the presented algorithm is linearly proportional to the number of triangles in the surface mesh of the surface. We showcase the usage of the algorithm with various examples of moving domain problems. We further extend the notion of universal meshes to generate a conforming triangulation for a given smooth curve from a generalized anisotropic background mesh. In particular method allows for triangles with obtuse angles in the background mesh. The novelty of the approach lies in the way the set of faces are selected to discretize the curve, and in the projection map constructed for the selected faces that discretize the given curve. We discuss two particular cases where the proposed method fails to generate conforming meshes. We generate these conforming meshes without altering the connectivity of the background mesh. Furthermore, we present preliminary results of using the same method for continuous curves, and anisotropic tetrahedral meshes for three-dimensional geometries. The universal meshes algorithm presented here is advantageous for a few important reasons. The method does not require that any global problem be solved; the background mesh is only generated once; and the connectivity of the background mesh is left unaltered. Hence, the computational complexity of mesh generation is observed to be proportional to the number of triangles on the surface. The examples and discussion presented here indicate that universal meshes can be a useful tool in simulating realistic engineering problems.
7. Fully implicit and semi-implicit hybrid discontinuous space-time Galerkin methods for acoustic wave propagation [electronic resource] [2014] Online
- Book
- 1 online resource.
This dissertation, based on the concept of the existing discontinuous Enrichment method (DEM) for frequency domain analysis, proposes a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions. This hybrid DGM extends concepts of DEM into the time domain for problems that are better suited for analysis in time domain. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this dissertation, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Numerical results for two and three dimensional model problems, in both low and mid frequency regimes, show that the proposed DGM outperforms the conventional space-time finite element method and Newmark family semi-discrete schemes. Additionally an alternative semi-implicit formulation is proposed where global level linear systems stemming from the implicit formulation is traded in favour of smaller and independent local systems. Numerical results for two dimensional model problems, in both low and mid frequency regimes, show that for a fixed mesh resolution, the semi-implicit DGM requires far less memory than its fully implicit counterpart. The semi-implicit scheme also parallelizes and scales very well with the number of available CPUs.
This dissertation, based on the concept of the existing discontinuous Enrichment method (DEM) for frequency domain analysis, proposes a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions. This hybrid DGM extends concepts of DEM into the time domain for problems that are better suited for analysis in time domain. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this dissertation, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Numerical results for two and three dimensional model problems, in both low and mid frequency regimes, show that the proposed DGM outperforms the conventional space-time finite element method and Newmark family semi-discrete schemes. Additionally an alternative semi-implicit formulation is proposed where global level linear systems stemming from the implicit formulation is traded in favour of smaller and independent local systems. Numerical results for two dimensional model problems, in both low and mid frequency regimes, show that for a fixed mesh resolution, the semi-implicit DGM requires far less memory than its fully implicit counterpart. The semi-implicit scheme also parallelizes and scales very well with the number of available CPUs.
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8. Implicit and higher-order discretization methods for compressible multi-phase fluid and fluid-structure problems [electronic resource] [2014] Online
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- 1 online resource.
This thesis describes an embedded boundary framework for multi-phase flow and fluid-structure interaction. This is based on the FIVER method, first introduced in (Farhat, 2008). This framework is capable of handling complex, evolving material interfaces, including structural fracture. This thesis makes a variety of contributions. Among the major contributions are an implicit time stepping method for embedded boundaries with highly nonlinear physical phenomena, which is shown to be up to 40x faster than the corresponding explicit time stepping scheme. Also, a multigrid method for the FIVER embedded boundary method is presented, which enables the solution of steady flow problems nearly 12x faster than a standard Newton-Krylov time stepping scheme. Then a second order extension of FIVER is introduced for multi-fluid and fluid-structure interaction problems. Finally, the stability and accuracy properties of this extension are analyzed.
This thesis describes an embedded boundary framework for multi-phase flow and fluid-structure interaction. This is based on the FIVER method, first introduced in (Farhat, 2008). This framework is capable of handling complex, evolving material interfaces, including structural fracture. This thesis makes a variety of contributions. Among the major contributions are an implicit time stepping method for embedded boundaries with highly nonlinear physical phenomena, which is shown to be up to 40x faster than the corresponding explicit time stepping scheme. Also, a multigrid method for the FIVER embedded boundary method is presented, which enables the solution of steady flow problems nearly 12x faster than a standard Newton-Krylov time stepping scheme. Then a second order extension of FIVER is introduced for multi-fluid and fluid-structure interaction problems. Finally, the stability and accuracy properties of this extension are analyzed.
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9. Topology optimization of a MEMS device [electronic resource] [2013] Online
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- 1 online resource.
The mechanics of microelectromechanical systems (MEMS) are typically designed using a set of popular building blocks (rectangular cantilevers, folded flexure beams, crab legs etc.). Optimization of these structures or their variants is often performed using scaling laws, parametric optimization, or some insight gained from studying relationships between certain features and a device's behavior. Topology optimization is a more powerful tool that systematically generates the full topology of a design, including the size, shape, and location of features, and can satisfy several goals despite potentially complex relationships. The focus of this thesis is to answer the following question: Is it advantageous to design MEMS structures such as RF MEMS capacitive switches using topology optimization? This thesis takes the reader through a full design process. The problem setup and problem formulation are justified in depth. The mechanical behaviors of devices with stresses and stress gradients resulting from typical micro-fabrication processes are explained. The finite element simulations are described, and the modeling decisions that can be particularly relevant to other MEMS problems are highlighted. The topology optimization method is thoroughly explained, and the challenges and necessary adaptations to the method are exposed. Sets of topology optimized designs are presented; guidelines for future intuitive design are extracted from an examination of the resulting geometries. Experimental data is provided, justifying many of the decisions taken during the design process, and validating the finite element models and topology optimization results. The experimental results also provide supplemental understanding regarding capacitive switch mechanics. The new knowledge should be integrated into any future problem formulation. We conclude that topology optimization can be used for MEMS design, significantly increasing the design possibilities and solving complex, non-intuitive problems.
The mechanics of microelectromechanical systems (MEMS) are typically designed using a set of popular building blocks (rectangular cantilevers, folded flexure beams, crab legs etc.). Optimization of these structures or their variants is often performed using scaling laws, parametric optimization, or some insight gained from studying relationships between certain features and a device's behavior. Topology optimization is a more powerful tool that systematically generates the full topology of a design, including the size, shape, and location of features, and can satisfy several goals despite potentially complex relationships. The focus of this thesis is to answer the following question: Is it advantageous to design MEMS structures such as RF MEMS capacitive switches using topology optimization? This thesis takes the reader through a full design process. The problem setup and problem formulation are justified in depth. The mechanical behaviors of devices with stresses and stress gradients resulting from typical micro-fabrication processes are explained. The finite element simulations are described, and the modeling decisions that can be particularly relevant to other MEMS problems are highlighted. The topology optimization method is thoroughly explained, and the challenges and necessary adaptations to the method are exposed. Sets of topology optimized designs are presented; guidelines for future intuitive design are extracted from an examination of the resulting geometries. Experimental data is provided, justifying many of the decisions taken during the design process, and validating the finite element models and topology optimization results. The experimental results also provide supplemental understanding regarding capacitive switch mechanics. The new knowledge should be integrated into any future problem formulation. We conclude that topology optimization can be used for MEMS design, significantly increasing the design possibilities and solving complex, non-intuitive problems.
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10. Algorithmic and experimental contributions to the study of ballistic penetration of tissue-like materials [electronic resource] [2012] Online
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- 1 online resource.
In this work we present a series of scientific contributions made to the study of the impact of projectiles into tissue-like materials, specifically the synthetic artificial tissue simulant Perma-Gel. These contributions consist of a combination of experimental observations, algorithmic ideas and numerical tools which demonstrate a series of problems and solutions to trying to simulate nearly incompressible soft tissues using finite elements. A number of experiments were performed by taking high-speed footage of the firing of spherical steel bullets at different speeds into Perma-Gel, a new thermoplastic material used as a proxy to human muscle tissue. This work appears to be the first publicly released experimental work using Perma-Gel and is part of the small amount of non-classified work looking at ballistic gelatin behavior. A number of experimental observations were made regarding the material behavior, elastic and plastic deformation around the projectile, and the possibility of cavitation. This work introduces an explicit dynamic contact algorithm that takes advantage of the asynchronous time stepping nature of Asynchronous Variational Integrators (AVI) to improve performance when simulating elastic-body rigid-wall contact. We demonstrate a number of desirable properties over traditional one-time-step methods for the simulation of solid dynamics and provide a number of examples highlighting the advantages of this method. The explicit contact algorithm and AVI was used to simulate the impact of a projectile into a simulated block of gelatin, but was hindered by difficulties using the realistic material parameters. Using a parallelized version of the algorithm, large-scale simulations were performed for progressively smaller shear moduli. As the simulations approached realistic values for the shear modulus, unstable element configurations formed which required infeasibly small time steps to successfully resolve. The behavior observed for the shear moduli we could numerically simulate with did not resemble the experimental results. To simulate with smaller values, we had to go to an axisymmetric setting. The axisymmetric setting increased the range of shear moduli which could be simulated and demonstrated the same dynamic behavior, though the issue of unstable element configurations continued to occur in extreme cases. To deal with the issue of unstable elements, we created an axisymmetric remeshing strategy to compensate for the unstable element configurations and insufficient spatial resolution. This strategy consists of periodically applying a remeshing and transfer algorithm that updates highly deformed finite element meshes with configurations formed with elements having uniform aspect ratios and local refinement in important areas. The axisymmetric setting with remeshing increased the range of potential shear modulus values that could be simulated. This allowed for the identification of qualitative similarities in the transient behavior between the numerical results and the experimental footage.
In this work we present a series of scientific contributions made to the study of the impact of projectiles into tissue-like materials, specifically the synthetic artificial tissue simulant Perma-Gel. These contributions consist of a combination of experimental observations, algorithmic ideas and numerical tools which demonstrate a series of problems and solutions to trying to simulate nearly incompressible soft tissues using finite elements. A number of experiments were performed by taking high-speed footage of the firing of spherical steel bullets at different speeds into Perma-Gel, a new thermoplastic material used as a proxy to human muscle tissue. This work appears to be the first publicly released experimental work using Perma-Gel and is part of the small amount of non-classified work looking at ballistic gelatin behavior. A number of experimental observations were made regarding the material behavior, elastic and plastic deformation around the projectile, and the possibility of cavitation. This work introduces an explicit dynamic contact algorithm that takes advantage of the asynchronous time stepping nature of Asynchronous Variational Integrators (AVI) to improve performance when simulating elastic-body rigid-wall contact. We demonstrate a number of desirable properties over traditional one-time-step methods for the simulation of solid dynamics and provide a number of examples highlighting the advantages of this method. The explicit contact algorithm and AVI was used to simulate the impact of a projectile into a simulated block of gelatin, but was hindered by difficulties using the realistic material parameters. Using a parallelized version of the algorithm, large-scale simulations were performed for progressively smaller shear moduli. As the simulations approached realistic values for the shear modulus, unstable element configurations formed which required infeasibly small time steps to successfully resolve. The behavior observed for the shear moduli we could numerically simulate with did not resemble the experimental results. To simulate with smaller values, we had to go to an axisymmetric setting. The axisymmetric setting increased the range of shear moduli which could be simulated and demonstrated the same dynamic behavior, though the issue of unstable element configurations continued to occur in extreme cases. To deal with the issue of unstable elements, we created an axisymmetric remeshing strategy to compensate for the unstable element configurations and insufficient spatial resolution. This strategy consists of periodically applying a remeshing and transfer algorithm that updates highly deformed finite element meshes with configurations formed with elements having uniform aspect ratios and local refinement in important areas. The axisymmetric setting with remeshing increased the range of potential shear modulus values that could be simulated. This allowed for the identification of qualitative similarities in the transient behavior between the numerical results and the experimental footage.
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11. Design of bio-inspired directional tapered adhesives and hierarchies [electronic resource] [2012] Online
- Book
- 1 online resource.
Research into how the gecko lizard is able to climb a wide variety of surfaces has re- vealed an adhesive system that takes a fundamentally different approach than is found in conventional pressure-sensitive adhesives such as sticky tape. The gecko's adhesive system is composed of setal stalks, each thinner than a human hair and terminating in spatulae only 250 nm across. The entire hierarchical system is composed of beta- keratin, a tough, hydrophobic material, somewhat harder than the alpha-keratin of human fingernails. The geometry of the setae and spatulae allow them to conform to surfaces in a manner similar to very soft materials, but without the tendency of tacky materials to become fouled with dirt. Using the gecko adhesive system as inspiration, Biomimetics and Dexterous Ma- nipulation Laboratory developed an adhesive that is suitable for robotic climbing ap- plications. The smallest features of this adhesive are arrays of sharp wedges molded from silicone rubber. A tapered feature was pursued because it is capable of repro- ducing the "frictional adhesion" property of the gecko's adhesive system. Frictional- adhesion defines a behavior for which increasing the shear stress imposed at a contact increases the available adhesive stress perpendicular to the surface. A consequence of frictional adhesion is that one can control the amount of adhesion by controlling the applied shear load. In the present case, the behavior arises from the fact that sharp wedge-shaped features initially present very little area as they are brought into contact with a surface. However, they bend over when the array is loaded in shear, so that the contact area and the adhesion grow in proportion. This thesis seeks to understand how the details of the tapered wedge geometry, including the wedge profile and angle of inclination, influence the frictional adhesive behavior. The analysis includes a combination of numerical finite element modeling and empirical pull-off tests. The constraints on material stiffness, wedge geometry and spacing are also studied, as affected by possible failure modes such as self-sticking of adjacent wedges (leading to "clumping"). The desire to test wedges at various angles of inclination lead to the development of a new micro-machining process for creating molds for the wedge arrays. This process affords much greater freedom to control the wedge size and geometry than a previous lithographic process. However, a byproduct of the machining process is that the wedges have a non-negligible surface roughness on their contacting faces, which compromises their performance. Consequently, a new process was developed to improve the surface finish by "inking" the molded wedges, depositing a thin film of liquid silicone rubber onto their faces and providing a smoother surface. The resulting microwedges achieve more than double the maximum adhesion and several times the adhesion at low levels of shear than previous microwedges from molds created using the lithographic process. Although the microwedges stick well to smooth, flat surfaces such as glass, they cannot conform to surfaces with undulations higher than a couple of micrometers. In addition, the array of microwedges must be precisely aligned with surfaces so that all wedges are uniformly loaded. To mitigate these limitations, some approximation to the gecko's compliant hierarchy of lamellae, setae and spatulae is needed. The solution presented in this thesis is a two-layer hierarchical system in which the arrays of wedges are supported by a larger array of angled pillars. In between the pillars and wedges is a film of solid silicone rubber, which bridges the gaps between pillars and helps to create a relatively uniform loading of the wedges. A combination of numerical analysis and empirical pull-off tests is used to understand the relationships among pillar dimensions, pillar spacing and film thickness that govern the performance of this structure. At one extreme, the loading can become sufficiently non-uniform that some wedges lose contact with the surface, resulting in a loss of adhesion. At the other extreme, the structure is too stiff to accommodate surface undulations and misalignment. The thesis concludes with a summary of the results on wedges and hierarchical adhesive structures, and discusses the implications for future work.
Research into how the gecko lizard is able to climb a wide variety of surfaces has re- vealed an adhesive system that takes a fundamentally different approach than is found in conventional pressure-sensitive adhesives such as sticky tape. The gecko's adhesive system is composed of setal stalks, each thinner than a human hair and terminating in spatulae only 250 nm across. The entire hierarchical system is composed of beta- keratin, a tough, hydrophobic material, somewhat harder than the alpha-keratin of human fingernails. The geometry of the setae and spatulae allow them to conform to surfaces in a manner similar to very soft materials, but without the tendency of tacky materials to become fouled with dirt. Using the gecko adhesive system as inspiration, Biomimetics and Dexterous Ma- nipulation Laboratory developed an adhesive that is suitable for robotic climbing ap- plications. The smallest features of this adhesive are arrays of sharp wedges molded from silicone rubber. A tapered feature was pursued because it is capable of repro- ducing the "frictional adhesion" property of the gecko's adhesive system. Frictional- adhesion defines a behavior for which increasing the shear stress imposed at a contact increases the available adhesive stress perpendicular to the surface. A consequence of frictional adhesion is that one can control the amount of adhesion by controlling the applied shear load. In the present case, the behavior arises from the fact that sharp wedge-shaped features initially present very little area as they are brought into contact with a surface. However, they bend over when the array is loaded in shear, so that the contact area and the adhesion grow in proportion. This thesis seeks to understand how the details of the tapered wedge geometry, including the wedge profile and angle of inclination, influence the frictional adhesive behavior. The analysis includes a combination of numerical finite element modeling and empirical pull-off tests. The constraints on material stiffness, wedge geometry and spacing are also studied, as affected by possible failure modes such as self-sticking of adjacent wedges (leading to "clumping"). The desire to test wedges at various angles of inclination lead to the development of a new micro-machining process for creating molds for the wedge arrays. This process affords much greater freedom to control the wedge size and geometry than a previous lithographic process. However, a byproduct of the machining process is that the wedges have a non-negligible surface roughness on their contacting faces, which compromises their performance. Consequently, a new process was developed to improve the surface finish by "inking" the molded wedges, depositing a thin film of liquid silicone rubber onto their faces and providing a smoother surface. The resulting microwedges achieve more than double the maximum adhesion and several times the adhesion at low levels of shear than previous microwedges from molds created using the lithographic process. Although the microwedges stick well to smooth, flat surfaces such as glass, they cannot conform to surfaces with undulations higher than a couple of micrometers. In addition, the array of microwedges must be precisely aligned with surfaces so that all wedges are uniformly loaded. To mitigate these limitations, some approximation to the gecko's compliant hierarchy of lamellae, setae and spatulae is needed. The solution presented in this thesis is a two-layer hierarchical system in which the arrays of wedges are supported by a larger array of angled pillars. In between the pillars and wedges is a film of solid silicone rubber, which bridges the gaps between pillars and helps to create a relatively uniform loading of the wedges. A combination of numerical analysis and empirical pull-off tests is used to understand the relationships among pillar dimensions, pillar spacing and film thickness that govern the performance of this structure. At one extreme, the loading can become sufficiently non-uniform that some wedges lose contact with the surface, resulting in a loss of adhesion. At the other extreme, the structure is too stiff to accommodate surface undulations and misalignment. The thesis concludes with a summary of the results on wedges and hierarchical adhesive structures, and discusses the implications for future work.
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12. A finite element approach to the simulation of hydraulic fractures with lag [electronic resource] [2012] Online
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In this thesis, a finite-element-based algorithm is presented to simulate plane-strain hydraulic fractures in an impermeable elastic medium where the crack path is not known a priori. The algorithm accounts for the nonlinear coupling between the fluid pressure and the crack opening and separately tracks the evolution of the crack tip and the fluid front. It therefore allows the existence of a fluid lag. The fluid front is advanced explicitly in time, but the crack tip is determined implicitly by enforcing Griffith's criterion and maximum energy release rate. A spatial discretization is created that conforms to the crack path by perturbing the nodes of a background mesh. The coupling between the fluid and the rock is enforced by simultaneously solving for the fluid pressure and the crack opening at each time step. Verification of the algorithm is provided for straight hydraulic fractures by performing sample simulations and comparing them to two known similarity solutions. Also, sample simulations are carried out for the general case of curvilinear fractures for which the crack path is not known a priori.
In this thesis, a finite-element-based algorithm is presented to simulate plane-strain hydraulic fractures in an impermeable elastic medium where the crack path is not known a priori. The algorithm accounts for the nonlinear coupling between the fluid pressure and the crack opening and separately tracks the evolution of the crack tip and the fluid front. It therefore allows the existence of a fluid lag. The fluid front is advanced explicitly in time, but the crack tip is determined implicitly by enforcing Griffith's criterion and maximum energy release rate. A spatial discretization is created that conforms to the crack path by perturbing the nodes of a background mesh. The coupling between the fluid and the rock is enforced by simultaneously solving for the fluid pressure and the crack opening at each time step. Verification of the algorithm is provided for straight hydraulic fractures by performing sample simulations and comparing them to two known similarity solutions. Also, sample simulations are carried out for the general case of curvilinear fractures for which the crack path is not known a priori.
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13. High-order discontinuous Galerkin method with Lagrange multipliers for advection-diffusion problems [electronic resource] [2012] Online
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- 1 online resource.
A high-order discontinuous Galerkin method with Lagrange multipliers (DGLM) is proposed for solving the advection-diffusion equation on unstructured meshes. Following the same methodology as the discontinuous enrichment method (DEM), free-space solutions of the governing homogeneous partial differential equation are used in lieu of standard polynomials in order to finely capture features of the solution. Polynomial Lagrange multipliers are used to enforce weak continuity of the solution at the element interfaces. The design of arbitrary-order elements for homogeneous problems is discussed in detail and is supported by a mathematical analysis. For solving non-homogeneous advection-diffusion problems, a novel approach is proposed to decrease the computational cost. Adaptivity of the method is highlighted by the use of an a posteriori error estimate for automatic mesh refinement. The numerical results reveal that these DGLM elements outperform their standard Galerkin and stabilized Galerkin counterparts of comparable computational complexity.
A high-order discontinuous Galerkin method with Lagrange multipliers (DGLM) is proposed for solving the advection-diffusion equation on unstructured meshes. Following the same methodology as the discontinuous enrichment method (DEM), free-space solutions of the governing homogeneous partial differential equation are used in lieu of standard polynomials in order to finely capture features of the solution. Polynomial Lagrange multipliers are used to enforce weak continuity of the solution at the element interfaces. The design of arbitrary-order elements for homogeneous problems is discussed in detail and is supported by a mathematical analysis. For solving non-homogeneous advection-diffusion problems, a novel approach is proposed to decrease the computational cost. Adaptivity of the method is highlighted by the use of an a posteriori error estimate for automatic mesh refinement. The numerical results reveal that these DGLM elements outperform their standard Galerkin and stabilized Galerkin counterparts of comparable computational complexity.
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3781 2012 B | In-library use |
14. High-order embedded boundary methods for fluid-structure interaction [electronic resource] [2012] Online
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- 1 online resource.
Embedded boundary methods are gaining popularity for solving fluid-structure interaction (FSI) problems because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid-structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, embedded boundary methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid-structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the embedded boundaries. In some cases, they are also provably inconsistent at these locations. To address this issue, the present work first presents a systematic approach for constructing higher-order embedded boundary methods for fluid-structure interaction problems. This approach is developed for inviscid flows and fluid-structure interaction problems. For the sake of clarity, but without any loss of generality, the methodology is described in one and two dimensions. However, its extension to three-dimensional problems is straightforward and illustrated by numerical examples. The proposed approach leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the order of spatial accuracy of this scheme. It is illustrated in this work by its application to various finite difference (FD) and finite volume (FV) methods. Its impact is demonstrated by numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces. Next, attention is focused on a limiter issue that is associated with second-order FV methods. In most practical higher-resolution FV methods, like for example second-order MUSCL schemes, slope limiters are employed. They are designed to suppress spurious oscillations near discontinuities as well as preserve second-order accuracy in the region where the solution is sufficiently smooth. Most slope limiter functions are developed assuming one-dimensional uniform CFD grids however, are applied as such in practice to unstructured meshes. It is observed that for a cell-centered FV scheme, these conventional limiter functions lead to loss of spatial accuracy when the fluid meshes are non-uniform. To fix this issue, the Reconstruction-Evolve-Project (REP) procedure is generalized to account for non-uniform grids. The effect of slope limiters on accuracy and total- variational-diminishing (TVD) stability is also studied. A series of mathematical conditions that the slope limiters must satisfy in order to deliver desired numerical properties are derived. Several most widely used conventional slope limiter functions are enhanced to satisfy these conditions. The impact of the enhanced slope limiter functions is demonstrated by solving benchmark problems on highly non-uniform CFD meshes that confirm: (1) the ability to maintain second-order accuracy in space and (2) the ability to suppress spurious oscillations near discontinuities.
Embedded boundary methods are gaining popularity for solving fluid-structure interaction (FSI) problems because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid-structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, embedded boundary methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid-structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the embedded boundaries. In some cases, they are also provably inconsistent at these locations. To address this issue, the present work first presents a systematic approach for constructing higher-order embedded boundary methods for fluid-structure interaction problems. This approach is developed for inviscid flows and fluid-structure interaction problems. For the sake of clarity, but without any loss of generality, the methodology is described in one and two dimensions. However, its extension to three-dimensional problems is straightforward and illustrated by numerical examples. The proposed approach leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the order of spatial accuracy of this scheme. It is illustrated in this work by its application to various finite difference (FD) and finite volume (FV) methods. Its impact is demonstrated by numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces. Next, attention is focused on a limiter issue that is associated with second-order FV methods. In most practical higher-resolution FV methods, like for example second-order MUSCL schemes, slope limiters are employed. They are designed to suppress spurious oscillations near discontinuities as well as preserve second-order accuracy in the region where the solution is sufficiently smooth. Most slope limiter functions are developed assuming one-dimensional uniform CFD grids however, are applied as such in practice to unstructured meshes. It is observed that for a cell-centered FV scheme, these conventional limiter functions lead to loss of spatial accuracy when the fluid meshes are non-uniform. To fix this issue, the Reconstruction-Evolve-Project (REP) procedure is generalized to account for non-uniform grids. The effect of slope limiters on accuracy and total- variational-diminishing (TVD) stability is also studied. A series of mathematical conditions that the slope limiters must satisfy in order to deliver desired numerical properties are derived. Several most widely used conventional slope limiter functions are enhanced to satisfy these conditions. The impact of the enhanced slope limiter functions is demonstrated by solving benchmark problems on highly non-uniform CFD meshes that confirm: (1) the ability to maintain second-order accuracy in space and (2) the ability to suppress spurious oscillations near discontinuities.
15. Universal meshes, a new paradigm for computing with nonconforming triangulations [electronic resource] [2012] Online
- Book
- 1 online resource.
Finite element methods commonly handle evolving domains in one of two ways-- either the changing domain is remeshed at each instant/update, or it is immersed in a background mesh and approximated within it. We introduce a novel approach here that inherits the conceptual simplicity of the former and the computational efficiency of the latter. We describe a method for exactly discretizing planar C2-regular domains immersed in nonconforming triangulations. The key idea in discretizing curved domains is to map triangles in a background mesh to curvilinear ones that conform exactly with the boundary. We construct such mappings using a novel way of parameterizing a curved boundary over a nearby collections of edges with its closest point projection. Then, extending this parameterization to a small neighborhood of the boundary in a piecewise smooth manner yields a discretization for the domain itself. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the background mesh. Indeed, interpolating just at the vertices of the background mesh yields a fast meshing algorithm that involves only perturbing vertices near the boundary. For our method of discretizing of a curved domain to be robust, we have to impose restrictions on the background mesh. Conversely, these restrictions define a family of domains that can be discretized with a given background mesh. We then say that the background mesh is a universal mesh for such a family of domains. The notion of universal meshes is particularly useful in free/moving boundary problems because the same background mesh can serve as the universal mesh for an evolving domain for time intervals independent of the time step. Hence, it facilitates a framework for finite element calculations over evolving domains while using a fixed background mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. The main challenge in our method is determining when a domain can be discretized using a given background mesh. In turn, this depends on when the parameterization that we compute for its boundary is robust. To this end, we identify sufficient conditions under which the restriction of the closest point projection to the selected edges is a homeomorphism onto the boundary. Specifically, we require that the background mesh be sufficiently refined and that certain interior angles of its triangles near the boundary be strictly acute. We provide local computable estimates for the required mesh size and for the Jacobian of the resulting parameterization. We show that the latter is bounded and positive independent of the mesh size. These assumptions, along with a possibly smaller mesh size, also guarantee that the method for discretizing curved domains is robust. Three factors are pivotal to the success of the idea of universal meshes for simulating free/moving boundary problems. Firstly, there are no conformity requirements on the background mesh; none of its vertices need to lie on the boundary. In fact, a sufficiently refined mesh of equilateral triangles suffices to discretize any smooth planar boundary/domain. Secondly, the restrictions we do impose on background meshes can be both easily satisfied and checked. Finally, by using same background mesh to simulate evolving geometries for (reasonably) long times, it is possible to retain the sparsity patterns of data structures involved in the problem. We present numerous examples to demonstrate the high-order of convergence possible with the discretization method. We include simulations of flows over domains with moving boundaries. We are also investigating applications to simulating the propagation of curved cracks and the dynamics of fluid membranes. These examples and applications indicate that universal meshes can be a useful tool in simulating a challenging class of problems in realistic engineering applications.
Finite element methods commonly handle evolving domains in one of two ways-- either the changing domain is remeshed at each instant/update, or it is immersed in a background mesh and approximated within it. We introduce a novel approach here that inherits the conceptual simplicity of the former and the computational efficiency of the latter. We describe a method for exactly discretizing planar C2-regular domains immersed in nonconforming triangulations. The key idea in discretizing curved domains is to map triangles in a background mesh to curvilinear ones that conform exactly with the boundary. We construct such mappings using a novel way of parameterizing a curved boundary over a nearby collections of edges with its closest point projection. Then, extending this parameterization to a small neighborhood of the boundary in a piecewise smooth manner yields a discretization for the domain itself. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the background mesh. Indeed, interpolating just at the vertices of the background mesh yields a fast meshing algorithm that involves only perturbing vertices near the boundary. For our method of discretizing of a curved domain to be robust, we have to impose restrictions on the background mesh. Conversely, these restrictions define a family of domains that can be discretized with a given background mesh. We then say that the background mesh is a universal mesh for such a family of domains. The notion of universal meshes is particularly useful in free/moving boundary problems because the same background mesh can serve as the universal mesh for an evolving domain for time intervals independent of the time step. Hence, it facilitates a framework for finite element calculations over evolving domains while using a fixed background mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. The main challenge in our method is determining when a domain can be discretized using a given background mesh. In turn, this depends on when the parameterization that we compute for its boundary is robust. To this end, we identify sufficient conditions under which the restriction of the closest point projection to the selected edges is a homeomorphism onto the boundary. Specifically, we require that the background mesh be sufficiently refined and that certain interior angles of its triangles near the boundary be strictly acute. We provide local computable estimates for the required mesh size and for the Jacobian of the resulting parameterization. We show that the latter is bounded and positive independent of the mesh size. These assumptions, along with a possibly smaller mesh size, also guarantee that the method for discretizing curved domains is robust. Three factors are pivotal to the success of the idea of universal meshes for simulating free/moving boundary problems. Firstly, there are no conformity requirements on the background mesh; none of its vertices need to lie on the boundary. In fact, a sufficiently refined mesh of equilateral triangles suffices to discretize any smooth planar boundary/domain. Secondly, the restrictions we do impose on background meshes can be both easily satisfied and checked. Finally, by using same background mesh to simulate evolving geometries for (reasonably) long times, it is possible to retain the sparsity patterns of data structures involved in the problem. We present numerous examples to demonstrate the high-order of convergence possible with the discretization method. We include simulations of flows over domains with moving boundaries. We are also investigating applications to simulating the propagation of curved cracks and the dynamics of fluid membranes. These examples and applications indicate that universal meshes can be a useful tool in simulating a challenging class of problems in realistic engineering applications.
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16. A computational framework based on an embedded boundary method for nonlinear multi-phase fluid-structure interactions [electronic resource] [2011] Online
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- 1 online resource.
Nonlinear fluid-structure interaction (FSI) is a dominating feature in many important engineering applications. Examples include underwater implosions, pipeline explosions, flapping wings for micro aerial vehicles, and shock wave lithotripsy. Due to the inherent nonlinearity and system complexity, such problems have not been thoroughly analyzed, which greatly hinders the advance of related engineering fields. This thesis focuses on the development, verification, and validation of a fluid-structure coupled computational framework for the solution of nonlinear multi-phase FSI problems involving high compressions and shock waves, large structural displacements and deformations, self-contact, and possibly the initiation and propagation of cracks in the structure. First, an embedded boundary method for solving 3D multi-phase compressible inviscid flows on arbitrary (i.e. structured and unstructured) non body-conforming CFD grids is presented. Key components include: (1) robust and efficient computational algorithms for tracking open, closed, and cracking fluid-structure interfaces with respect to the fixed, non body-conforming CFD grid; (2) a numerical algorithm based on the exact solution of local, one-dimensional fluid-structure Riemann problems to enforce the no-interpenetration transmission condition at the fluid-structure interface; and (3) two consistent and conservative algorithms for enforcing the equilibrium transmission condition at the same interface. Next, the multi-phase compressible flow solver equipped with the aforementioned embedded boundary method is carefully coupled with an extended finite element method (XFEM) based structure solver, using a partitioned procedure and provably second-order explicit-explicit and implicit-explicit time-integrators. In particular, the interface tracking algorithms in the embedded boundary method are adapted to tracking embedded discrete interfaces with phantom elements and carrying implicitly represented cracks. Finally, the resulting fluid-structure coupled computational framework is applied to the solution of several challenging FSI problems in the fields of aeronautics, underwater implosions and explosions, and pipeline explosions to assess its performance. In particular, two laboratory experiments are considered for validation purpose: the first one concerns the implosive collapse of an air-filled aluminum cylinder; the second one studies the dynamic fracture of pre-flawed aluminum pipes driven by detonation waves. In both cases, the numerical simulation correctly reproduces in a quantitative sense the important features in the experiment.
Nonlinear fluid-structure interaction (FSI) is a dominating feature in many important engineering applications. Examples include underwater implosions, pipeline explosions, flapping wings for micro aerial vehicles, and shock wave lithotripsy. Due to the inherent nonlinearity and system complexity, such problems have not been thoroughly analyzed, which greatly hinders the advance of related engineering fields. This thesis focuses on the development, verification, and validation of a fluid-structure coupled computational framework for the solution of nonlinear multi-phase FSI problems involving high compressions and shock waves, large structural displacements and deformations, self-contact, and possibly the initiation and propagation of cracks in the structure. First, an embedded boundary method for solving 3D multi-phase compressible inviscid flows on arbitrary (i.e. structured and unstructured) non body-conforming CFD grids is presented. Key components include: (1) robust and efficient computational algorithms for tracking open, closed, and cracking fluid-structure interfaces with respect to the fixed, non body-conforming CFD grid; (2) a numerical algorithm based on the exact solution of local, one-dimensional fluid-structure Riemann problems to enforce the no-interpenetration transmission condition at the fluid-structure interface; and (3) two consistent and conservative algorithms for enforcing the equilibrium transmission condition at the same interface. Next, the multi-phase compressible flow solver equipped with the aforementioned embedded boundary method is carefully coupled with an extended finite element method (XFEM) based structure solver, using a partitioned procedure and provably second-order explicit-explicit and implicit-explicit time-integrators. In particular, the interface tracking algorithms in the embedded boundary method are adapted to tracking embedded discrete interfaces with phantom elements and carrying implicitly represented cracks. Finally, the resulting fluid-structure coupled computational framework is applied to the solution of several challenging FSI problems in the fields of aeronautics, underwater implosions and explosions, and pipeline explosions to assess its performance. In particular, two laboratory experiments are considered for validation purpose: the first one concerns the implosive collapse of an air-filled aluminum cylinder; the second one studies the dynamic fracture of pre-flawed aluminum pipes driven by detonation waves. In both cases, the numerical simulation correctly reproduces in a quantitative sense the important features in the experiment.
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17. Computational models of blood flow in deformable arteries incorporating viscoelastic wall properties [electronic resource] [2011] Online
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- 1 online resource.
It is well known that blood vessels exhibit viscoelastic properties. Vessel wall viscoelasticity is an important source of physical damping and dissipation in the cardiovascular system. There is a growing need to incorporate viscoelasticity of arteries in computational models of blood flow which are utilized for applications such as disease research, treatment planning and medical device evaluation. However, thus far the use of viscoelastic wall properties in blood flow modeling has been limited. As part of the present work, arterial wall viscoelasticity was incorporated into two computational models of blood flow: (1) a nonlinear one-dimensional (1-D) model and (2) a three-dimensional (3-D) fluid-solid interaction (FSI) model of blood flow. 1-D blood flow model: In blood flow simulations different viscoelastic wall models may produce significantly different flow, pressure and wall deformation solutions. To highlight these differences a novel comparative study of two viscoelastic wall models and an elastic model is presented in this work. The wall models were incorporated in a nonlinear 1-D model of blood flow, which was solved using a space-time finite element method. The comparative study involved the following applications: (i) Wave propagation in an idealized vessel with reflection-free outflow boundary condition; (ii) Carotid artery model with non-periodic boundary conditions; (iii) Subject-specific abdominal aorta model under rest and exercise conditions. 3-D FSI blood flow model: 3-D blood flow models enable physiologic simulations in complex, subject-specific anatomies. In the present work, a viscoelastic constitutive relationship for the arterial wall was incorporated in the 3-D Coupled Momentum Method for Fluid-Solid Interaction problems (CMM-FSI). Results in an idealized carotid artery stenosis geometry show that higher frequency components of flow rate, pressure and vessel wall motion are damped in the viscoelastic case. These results indicate that the dissipative nature of viscoelastic wall properties has an important impact in 3-D simulations of blood flow. Future work will include simulations of blood flow in patient-specific geometries such as aortic coarctation (a congenital disease) to assess the impact of wall viscoelasticity in clinically relevant scenarios. In the present work, arterial viscoelasticity has been incorporated in 1-D and 3-D computational models of blood flow. The biomechanical effects of wall viscoelasticity have been investigated through idealized and subject-specific blood flow simulations. These contributions are significant and suggest the potential importance of wall viscoelasticity in blood flow simulations for clinically relevant applications.
It is well known that blood vessels exhibit viscoelastic properties. Vessel wall viscoelasticity is an important source of physical damping and dissipation in the cardiovascular system. There is a growing need to incorporate viscoelasticity of arteries in computational models of blood flow which are utilized for applications such as disease research, treatment planning and medical device evaluation. However, thus far the use of viscoelastic wall properties in blood flow modeling has been limited. As part of the present work, arterial wall viscoelasticity was incorporated into two computational models of blood flow: (1) a nonlinear one-dimensional (1-D) model and (2) a three-dimensional (3-D) fluid-solid interaction (FSI) model of blood flow. 1-D blood flow model: In blood flow simulations different viscoelastic wall models may produce significantly different flow, pressure and wall deformation solutions. To highlight these differences a novel comparative study of two viscoelastic wall models and an elastic model is presented in this work. The wall models were incorporated in a nonlinear 1-D model of blood flow, which was solved using a space-time finite element method. The comparative study involved the following applications: (i) Wave propagation in an idealized vessel with reflection-free outflow boundary condition; (ii) Carotid artery model with non-periodic boundary conditions; (iii) Subject-specific abdominal aorta model under rest and exercise conditions. 3-D FSI blood flow model: 3-D blood flow models enable physiologic simulations in complex, subject-specific anatomies. In the present work, a viscoelastic constitutive relationship for the arterial wall was incorporated in the 3-D Coupled Momentum Method for Fluid-Solid Interaction problems (CMM-FSI). Results in an idealized carotid artery stenosis geometry show that higher frequency components of flow rate, pressure and vessel wall motion are damped in the viscoelastic case. These results indicate that the dissipative nature of viscoelastic wall properties has an important impact in 3-D simulations of blood flow. Future work will include simulations of blood flow in patient-specific geometries such as aortic coarctation (a congenital disease) to assess the impact of wall viscoelasticity in clinically relevant scenarios. In the present work, arterial viscoelasticity has been incorporated in 1-D and 3-D computational models of blood flow. The biomechanical effects of wall viscoelasticity have been investigated through idealized and subject-specific blood flow simulations. These contributions are significant and suggest the potential importance of wall viscoelasticity in blood flow simulations for clinically relevant applications.
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18. Mechanics of hysteretic adhesive elastic mechanical contact between rough surfaces [electronic resource] [2011] Online
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- 1 online resource.
In experiments that involve contact with adhesion between two surfaces, as found in atomic force microscopy or nanoindentation, two distinct contact force (P ) vs. indentation-depth (h) curves are often measured depending on whether the indenter moves towards or away from the sample. The origin of this hysteresis is not well understood and is often attributed to moisture, plasticity or viscoelasticity. We present experiments, atomistic simulations and continuum mechanics models that will show that hysteresis can exist without these effects, and that its magnitude depends on surface roughness. We explain the observed hysteresis as the result of a series of surface instabilities, where the contact area grows or recedes by a finite amount. We also demonstrate that when this is the case material properties can be estimated uniquely from contact experiments even when the measured P -h curves are not unique. The hysteresis energy loss during contact is also a measure of the adhesive toughness of the contact interface. We show experimentally that roughness can both increase and decrease the adhesive toughness of the contact interface. We show through numerical simulation of continuum adhesive contact models that the contact interface is optimally tough at conditions at which the contact region is at the cusp of the transition through which it turns form being mostly simply connected to being predominantly multiply connected.
In experiments that involve contact with adhesion between two surfaces, as found in atomic force microscopy or nanoindentation, two distinct contact force (P ) vs. indentation-depth (h) curves are often measured depending on whether the indenter moves towards or away from the sample. The origin of this hysteresis is not well understood and is often attributed to moisture, plasticity or viscoelasticity. We present experiments, atomistic simulations and continuum mechanics models that will show that hysteresis can exist without these effects, and that its magnitude depends on surface roughness. We explain the observed hysteresis as the result of a series of surface instabilities, where the contact area grows or recedes by a finite amount. We also demonstrate that when this is the case material properties can be estimated uniquely from contact experiments even when the measured P -h curves are not unique. The hysteresis energy loss during contact is also a measure of the adhesive toughness of the contact interface. We show experimentally that roughness can both increase and decrease the adhesive toughness of the contact interface. We show through numerical simulation of continuum adhesive contact models that the contact interface is optimally tough at conditions at which the contact region is at the cusp of the transition through which it turns form being mostly simply connected to being predominantly multiply connected.
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3781 2011 K | In-library use |
19. Time integrators based on approximate discontinuous hamiltonians [electronic resource] [2011] Online
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- 1 online resource.
In this work we present a framework with which to construct time integrators for general mechanical systems. By approximating the potential energy with a nearby one, one can define a class of time integrators with several desirable properties, such as unconditional stability and conservation of invariants of the motion to machine precision. The resulting algorithms are asynchronous, explicit and can very easily be extended to handle frictionless contact with rigid boundaries. We discuss how these ideas extend to finite element discretizations of nonlinear elastic bodies and propose and specialize these methods so they scale linearly with the number of degrees of freedom in the system. Initially, we consider piecewise constant and affine approximations to the exact potential energy, and numerical verify the convergence of the resulting integrators. It is seen that while the piecewise constant integrator conserves all invariants of the problem, it is only conditionally convergent. The piecewise affine approximation converges on all numerical experiments performed, but necessarily breaks rotational invariance and hence does not conserve angular momentum. We analyze the one-dimensional piecewise affine algorithm and prove that the convergence rate of the integrator in fact depends on whether or not the exact trajectory every reaches a ``turning point, '' defined as a point where the velocity is identically zero. If no such point is reached, the integrator converges in the trajectories and velocities like O(h^2), where h> 0 is a mesh parameter. If such a point is reached, a reduced convergence rate of O(h^{3/2}) is observed. The convergence proof uses Gronwall's inequality, so convergence follows from a summation of the local truncation errors, and proves the reduced convergence rate is due to a low order error incurred near such turning points. We conclude this dissertation by proposing an algorithm for finite element systems which overcomes the shortcomings of the piecewise constant and affine integrators, at the price of a higher computational cost. This idea is based on introducing a discontinuous displacement field at nodes, similar in spirit to a discontinuous Galerkin method, and defining a piecewise quadratic potential energy for each element. With this approach we define a material frame indifferent approximate potential energy, using the edge lengths of the element in the deformed configuration. We show herein that the equations of motion for each element completely decouple, so each element can be evolved independently of other elements and hence the algorithm is extremely amenable to parallelization. In light of this feature of the algorithm, we can address the high computational cost of the integrator with a CUDA implementation on a graphics processing unit, which greatly improves the algorithm's running time. The piecewise quadratic ADH algorithm converges numerically on all examples performed while still conserving all invariants of the original problem.
In this work we present a framework with which to construct time integrators for general mechanical systems. By approximating the potential energy with a nearby one, one can define a class of time integrators with several desirable properties, such as unconditional stability and conservation of invariants of the motion to machine precision. The resulting algorithms are asynchronous, explicit and can very easily be extended to handle frictionless contact with rigid boundaries. We discuss how these ideas extend to finite element discretizations of nonlinear elastic bodies and propose and specialize these methods so they scale linearly with the number of degrees of freedom in the system. Initially, we consider piecewise constant and affine approximations to the exact potential energy, and numerical verify the convergence of the resulting integrators. It is seen that while the piecewise constant integrator conserves all invariants of the problem, it is only conditionally convergent. The piecewise affine approximation converges on all numerical experiments performed, but necessarily breaks rotational invariance and hence does not conserve angular momentum. We analyze the one-dimensional piecewise affine algorithm and prove that the convergence rate of the integrator in fact depends on whether or not the exact trajectory every reaches a ``turning point, '' defined as a point where the velocity is identically zero. If no such point is reached, the integrator converges in the trajectories and velocities like O(h^2), where h> 0 is a mesh parameter. If such a point is reached, a reduced convergence rate of O(h^{3/2}) is observed. The convergence proof uses Gronwall's inequality, so convergence follows from a summation of the local truncation errors, and proves the reduced convergence rate is due to a low order error incurred near such turning points. We conclude this dissertation by proposing an algorithm for finite element systems which overcomes the shortcomings of the piecewise constant and affine integrators, at the price of a higher computational cost. This idea is based on introducing a discontinuous displacement field at nodes, similar in spirit to a discontinuous Galerkin method, and defining a piecewise quadratic potential energy for each element. With this approach we define a material frame indifferent approximate potential energy, using the edge lengths of the element in the deformed configuration. We show herein that the equations of motion for each element completely decouple, so each element can be evolved independently of other elements and hence the algorithm is extremely amenable to parallelization. In light of this feature of the algorithm, we can address the high computational cost of the integrator with a CUDA implementation on a graphics processing unit, which greatly improves the algorithm's running time. The piecewise quadratic ADH algorithm converges numerically on all examples performed while still conserving all invariants of the original problem.
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3781 2011 D | In-library use |
20. Time-parallel methods for accelerating the solution of structural dynamics problems [electronic resource] [2011] Online
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- 1 online resource.
The classical approach for solving evolution Partial Differential Equations (PDEs) using a parallel computer consists in first partitioning the spatial domain and assigning each subdomain to a processor to achieve space-parallelism, then advancing the solution sequentially. However, enabling parallelism along the time dimension, despite its intrinsic difficulty, can be of paramount importance to fast computations when space-parallelism is unfeasible, cannot fully exploit a massively parallel machine or when near-real-time prediction is desired. The aforementioned objective can be achieved by applying classical domain decomposition principles to the time axis. The latter is first partitioned into time-slices to be processed independently. Starting with approximate seed information that provides a set of initial conditions, the response is then advanced in parallel in each time-slice using a standard time-stepping integrator. This decomposed solution exhibits discontinuities or jumps at the time-slice boundaries if the initial guess is not accurate. Applying a Newton-like approach to the time-dependent system, a correction function is then computed to improve the accuracy of the seed values and the process is repeated until convergence is reached. Methods based on the above concept have been successfully applied to various problems but none was found to be competitive for even for the simplest of second-order hyperbolic PDEs, a class of equations that covers the field of structural dynamics among others. To overcome this difficulty, a key idea is to improve the sequential propagator used for correcting the seed values, observing that the original evolution problem and the derived corrective one are closely related. The present work first demonstrates how this insight can be brought to fruition in the context of linear oscillators, with numerical examples featuring structural models ranging from academic to more challenging large-scale ones. An extension of this method to nonlinear equations is then developed and its concrete application to geometrically nonlinear transient dynamics is presented. Finally, it is shown how the time-reversibility property that characterizes some of the above problems can be exploited to develop a new framework that provides an increased speed-up factor.
The classical approach for solving evolution Partial Differential Equations (PDEs) using a parallel computer consists in first partitioning the spatial domain and assigning each subdomain to a processor to achieve space-parallelism, then advancing the solution sequentially. However, enabling parallelism along the time dimension, despite its intrinsic difficulty, can be of paramount importance to fast computations when space-parallelism is unfeasible, cannot fully exploit a massively parallel machine or when near-real-time prediction is desired. The aforementioned objective can be achieved by applying classical domain decomposition principles to the time axis. The latter is first partitioned into time-slices to be processed independently. Starting with approximate seed information that provides a set of initial conditions, the response is then advanced in parallel in each time-slice using a standard time-stepping integrator. This decomposed solution exhibits discontinuities or jumps at the time-slice boundaries if the initial guess is not accurate. Applying a Newton-like approach to the time-dependent system, a correction function is then computed to improve the accuracy of the seed values and the process is repeated until convergence is reached. Methods based on the above concept have been successfully applied to various problems but none was found to be competitive for even for the simplest of second-order hyperbolic PDEs, a class of equations that covers the field of structural dynamics among others. To overcome this difficulty, a key idea is to improve the sequential propagator used for correcting the seed values, observing that the original evolution problem and the derived corrective one are closely related. The present work first demonstrates how this insight can be brought to fruition in the context of linear oscillators, with numerical examples featuring structural models ranging from academic to more challenging large-scale ones. An extension of this method to nonlinear equations is then developed and its concrete application to geometrically nonlinear transient dynamics is presented. Finally, it is shown how the time-reversibility property that characterizes some of the above problems can be exploited to develop a new framework that provides an increased speed-up factor.
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3781 2011 C | In-library use |
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