Computer Methods in Applied Mechanics and Engineering, 2022
We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement... more We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.
This paper proposes a computational methodology that allows a direct numerical simulation of hete... more This paper proposes a computational methodology that allows a direct numerical simulation of heterogeneous/functionally graded materials based on V-reps/V-models and the Finite Cell Method (FCM). The FCM is an embedded domain approach that employs higher-order finite elements. The basic idea is to embed a complex geometric model into a fictitious domain that is trivial to mesh. The complexity of the geometry is then recaptured by an adapted precise numerical integration scheme for the elements cut by the boundary. For this, only a robust point inclusion test is required, which can be provided by various Computer-Aided Design (CAD) models. V-rep is a geometric modeling framework that represents the entire volume based on tri-variate B-Splines. Consequently, not only a point inclusion test is provided – but also the possibility to represent and model the interior domain. This allows to apply functionally graded material based on the tri-variate basis functions. These material parameters...
Computer Methods in Applied Mechanics and Engineering, 2019
This paper proposes a computational methodology for the integration of Computer Aided Design (CAD... more This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside-outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also "dirty geometries", i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples.
Advanced Modeling and Simulation in Engineering Sciences, 2014
Background: It is a well-known fact that cross-laminated timber structures are sensitive to rumbl... more Background: It is a well-known fact that cross-laminated timber structures are sensitive to rumbling noises. These transmissions are best captured by a fully three-dimensional mathematical model. Since the discretization of such models with hexahedral elements in a conforming manner is highly complex, we chose the mortar method to reduce the algorithmic complexity for the mesh generation. Moreover we consider high-order finite elements in order to deal with the high aspect ratios in three-dimensionally resolved, cross-laminated walls and slabs. The geometric models and material specification was derived from a building information model. Methods: This paper derives a new mortar formulation designed to replace an explicitely discretized elastomer with a new coupling condition. To this end, tailored Robin conditions are applied at the interface as coupling conditions instead of the more standard continuity constraints. Having demonstrated the suitability of the mortar method for high order finite elements, we proceed with the derivation of the dimensional reduced model with the new coupling condition and to show its stability by numerical experiments. We then test the performance of the new formulation on benchmark examples and demonstrate the engineering relevance for practical applications. Results: The newly derived mortar formulation performs well. We tested the new formulation on fully three-dimensional examples of engineering relevance discretized by high-order finite elements up to degrees of p = 10 and found the reproduction of both eigenvalues and eigenmodes to be accurate. Moreover, the mortar method allows for a significant reduction in the algorithmic complexity of mesh generation while simultaneously reducing the overall computational effort. Conclusion: The newly derived modified mortar method for replacing an elastomer layer is not only an academically interesting variant but is capable of solving problems of practical importance in modal-analysis of cross-laminated timber structures.
Computers & Mathematics with Applications, 2015
Contact problems in solid mechanics are traditionally solved using the h-version of the finite el... more Contact problems in solid mechanics are traditionally solved using the h-version of the finite element method. The constraints are enforced along the surfaces of e.g. elastic bodies under consideration. Standard constraint algorithms include penalty methods, Lagrange multiplier methods and combinations thereof. For complex scenarios, a major part of the solution time is taken up by operations to identify points that come into contact. This paper presents a novel approach to model frictionless contact using high order finite elements. Here, we employ an especially designed material model that is inserted into two-respectively threedimensional regions surrounding contacting bodies. Contact constraints are thus enforced on the same manifold as the accompanying structural problem. The application of the current material formulation leads to a regularization of the Karush-Kuhn-Tucker conditions. Our formulation can be classified as a barrier-type method. Results are obtained for two-and three-dimensional problems, including a Hertzian contact problem. Comparisons to a commercial FEA package are provided. The proposed formulation works well for non-matching discretizations on adjacent contact interfaces and handles self-contact naturally. Since the non-penetrating conditions are solved in a physically consistent manner, there is no need for an explicit contact search.
International Journal of Structural Stability and Dynamics, 2015
In this contribution, the finite cell method (FCM) is applied to solve transient problems of line... more In this contribution, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semi-discrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A 1D benchmark problem is solved using FCM, illustrating the method's ability to solve problems of linear elastodynamics obtaining high rates of convergence. Furthermore, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to the results obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics, offering a high degree of accuracy at a moderate com...
Lecture Notes in Computational Science and Engineering
In this contribution the use of hexahedral elements for the structural simulation in a fluid stru... more In this contribution the use of hexahedral elements for the structural simulation in a fluid structure interaction framework is presented, resulting in a consistent kinematic and geometric description of the solid. In order to compensate the additional numerical effort of the three-dimensional approach, an anisotropic p-adaptive method for linear elastodynamic problems is proposed, resulting in a clearly higher efficiency and higher convergence rates than uniform p-extensions. Special emphasis is placed on the accurate transfer of loads considering the fluid discretization for computation of the surface load integrals. For a coupling with a cartesian grid based Lattice Boltzmann code it was found that oscillations in the interface tractions may excite higher structural modes possibly leading to a nonstable coupling behavior. A first remedy to this problem was a linear modal analysis of the structure, thus allowing to control the number of modes to be considered without disregarding bidirectional fluid structure interactions. Preliminary results are presented for the FSI benchmark configuration proposed in this book.
After a short discussion of recent discretization techniques for the lattice-Boltzmann equations ... more After a short discussion of recent discretization techniques for the lattice-Boltzmann equations we motivate and discuss some alternative approaches using implicit, nonuniform FD discretization and mesh refinement techniques. After presenting results of a stability analysis we use an implicit approach to simulate a boundary layer test problem. The numerical results compare well to the reference solution when using strongly refined meshes. Some basic ideas for a nonuniform mesh refinement (with non-cartesian mesh topology) are introduced using the standard discretization procedure of alternating collision and propagation.
International Journal for Numerical Methods in Engineering, 2014
We present a parameter-free domain sewing approach for low-as well as high-order finite elements.... more We present a parameter-free domain sewing approach for low-as well as high-order finite elements. Its final form contains only primal unknowns, i.e., the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem dependent parameters which require an estimation. The presented approach is symmetry-preserving, i.e. the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non-matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p-and B-spline versions of the finite cell method, an embedded domain method of high order, and compared to more classical methods such as the penalty method or Nitsche's method.
The development of flow instabilities due to high Reynolds number flow in artificial heart valve ... more The development of flow instabilities due to high Reynolds number flow in artificial heart valve geometries inducing high strain rates and stresses often leads to hemolysis and related highly undesired effects. Geometric and functional optimization of artificial heart valves is therefore mandatory. In addition to experimental work in this field it is meanwhile possible to obtain increasing insight into flow dynamics by computer simulation of refined model problems. After giving an introductory overview we report the results of the simulation of three-dimensional transient physiological flows in fixed geometries similar to a CarboMedics bileaflet heart valve at different opening angles. The visualization of emerging complicated flow patterns gives detailed information about the transient history of the systems dynamical stability. Stress analysis indicates temporal shear stress peaks even far away from walls. The mathematical approach used is the Lattice Boltzmann method. We obtained reasonable results for velocity and shear stress fields. The code is implemented on parallel hardware in order to decrease computation time. Finally, we discuss problems, shortcomings and possible extensions of our approach.
IOP Conference Series: Materials Science and Engineering, 2010
The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM,... more The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM, permits the effective solution of problems with very complex geometry, since it circumvents the computationally expensive mesh generation and guarantees exponential convergence rates for smooth problems. The present contribution deals with the coupling of the FCM approach, which has been applied so far only to linear elasticity, with established nonlinear finite element technology. First, it is shown that the standard p-FEM based FCM converges poorly in a nonlinear formulation, since the presence of discontinuities leads to oscillatory solution fields. It is then demonstrated that the essential ideas of FCM, i.e. exponential convergence at virtually no meshing cost, can be achieved in the geometrically nonlinear setting, if high-order Legendre shape functions are replaced by a hierarchically enriched B-spline patch.
International Journal for Numerical Methods in Engineering, 2009
Over the last decade the Lattice Boltzmann method, which was derived from the kinetic gas theory,... more Over the last decade the Lattice Boltzmann method, which was derived from the kinetic gas theory, has matured as an efficient approach for solving Navier-Stokes equations. The p-FEM approach has proved to be highly efficient for a variety of problems in the field of structural mechanics. Our goal is to investigate the validity and efficiency of coupling the two approaches to simulate transient bidirectional Fluid-Structure interaction problems with geometrically non-linear structural deflections. A benchmark configuration of self-induced large oscillations for a flag attached to a cylinder can be accurately and efficiently reproduced within this setting. We describe in detail the force evaluation techniques, displacement transfers and the algorithm used to couple these completely different solvers as well as the results, and compare them with a benchmark reference solution computed by a monolithic finite element approach.
International Journal for Numerical Methods in Engineering, 2012
The Finite Cell Method (FCM) is a fictitious domain approach based on hierarchical Ansatz spaces ... more The Finite Cell Method (FCM) is a fictitious domain approach based on hierarchical Ansatz spaces of higher order. The method avoids time-consuming and often error-prone mesh-generation and favorably exploits Cartesian grids to embed structures of complex geometry in a simple shaped computational domain thus shifting parts of the computational effort from mesh generation to the computation within the embedding finite cells of regular shape. This paper presents an effective integration approach for voxel-based models of linear elasticity that drastically reduces the computational effort on cell level. The applied strategy allows the pre-computation of an essential part of the cell matrices and vectors of higher order, representing stiffness and load, respectively. Several benchmark problems show the potential of the proposed method in particular for heterogeneous material properties as common in biomedical applications based on computer tomography scans. The applied strategy ensures a fast computation for time-critical simulations and even allows user-interactive simulations for models of moderate size at a high level of accuracy.
A three-dimensional model of the human tibia bone subject to realistic static physiological condi... more A three-dimensional model of the human tibia bone subject to realistic static physiological conditions was developed using the p-version of the ÿnite-element method. The p-version of the ÿnite-element method in combination with the blending-function method enabled the construction of elements displaying accurate representation of the domain's boundary. This allowed the modeling of complex structures like bones using few elements and circumvented the problems of complexity and inaccuracy that are typical of current commercially available ÿnite-element mesh generators. In these methods large p-elements are integrated with high-order functions, permitting the coupling of the mechanical properties of each integration point directly to a voxel matrix extracted from computerized tomographies. This allowed order of 1000 di erent properties to be assigned per element thereby rendering a highly heterogeneous bone model. The ÿnal objective of this bone model is to o ers a platform for the study medical devices and bone composition and behavior in fracture processes and in pathologies as osteoporosis. The results show realistic stresses and displacements of the human tibia under static loading. The stresses are distributed along the whole bone and more pronounced towards the periphery, where there is a larger concentration of cortical bone.
... to give reasonable results, when the solids velocity is small compared to the lattice distanc... more ... to give reasonable results, when the solids velocity is small compared to the lattice distance times ... goal was to test the suitability of LB methods for a coupled fluidstructure flow problem in ... presented here using the LB method gave reasonable results for the prediction of velocity ...
Computers & Mathematics with Applications, 2014
In the case of dominating convection, standard Bubnov-Galerkin finite elements are known to deliv... more In the case of dominating convection, standard Bubnov-Galerkin finite elements are known to deliver oscillating discrete solutions for the convection-diffusion equation. This paper demonstrates that increasing the polynomial degree (p-extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given Péclet number at the nodes of a mesh.
Computer Methods in Applied Mechanics and Engineering, 2022
We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement... more We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.
This paper proposes a computational methodology that allows a direct numerical simulation of hete... more This paper proposes a computational methodology that allows a direct numerical simulation of heterogeneous/functionally graded materials based on V-reps/V-models and the Finite Cell Method (FCM). The FCM is an embedded domain approach that employs higher-order finite elements. The basic idea is to embed a complex geometric model into a fictitious domain that is trivial to mesh. The complexity of the geometry is then recaptured by an adapted precise numerical integration scheme for the elements cut by the boundary. For this, only a robust point inclusion test is required, which can be provided by various Computer-Aided Design (CAD) models. V-rep is a geometric modeling framework that represents the entire volume based on tri-variate B-Splines. Consequently, not only a point inclusion test is provided – but also the possibility to represent and model the interior domain. This allows to apply functionally graded material based on the tri-variate basis functions. These material parameters...
Computer Methods in Applied Mechanics and Engineering, 2019
This paper proposes a computational methodology for the integration of Computer Aided Design (CAD... more This paper proposes a computational methodology for the integration of Computer Aided Design (CAD) and the Finite Cell Method (FCM) for models with "dirty geometries". FCM, being a fictitious domain approach based on higher order finite elements, embeds the physical model into a fictitious domain, which can be discretized without having to take into account the boundary of the physical domain. The true geometry is captured by a precise numerical integration of elements cut by the boundary. Thus, an effective Point Membership Classification algorithm that determines the inside-outside state of an integration point with respect to the physical domain is a core operation in FCM. To treat also "dirty geometries", i.e. imprecise or flawed geometric models, a combination of a segment-triangle intersection algorithm and a flood fill algorithm being insensitive to most CAD model flaws is proposed to identify the affiliation of the integration points. The present method thus allows direct computations on geometrically and topologically flawed models. The potential and merit for practical applications of the proposed method is demonstrated by several numerical examples.
Advanced Modeling and Simulation in Engineering Sciences, 2014
Background: It is a well-known fact that cross-laminated timber structures are sensitive to rumbl... more Background: It is a well-known fact that cross-laminated timber structures are sensitive to rumbling noises. These transmissions are best captured by a fully three-dimensional mathematical model. Since the discretization of such models with hexahedral elements in a conforming manner is highly complex, we chose the mortar method to reduce the algorithmic complexity for the mesh generation. Moreover we consider high-order finite elements in order to deal with the high aspect ratios in three-dimensionally resolved, cross-laminated walls and slabs. The geometric models and material specification was derived from a building information model. Methods: This paper derives a new mortar formulation designed to replace an explicitely discretized elastomer with a new coupling condition. To this end, tailored Robin conditions are applied at the interface as coupling conditions instead of the more standard continuity constraints. Having demonstrated the suitability of the mortar method for high order finite elements, we proceed with the derivation of the dimensional reduced model with the new coupling condition and to show its stability by numerical experiments. We then test the performance of the new formulation on benchmark examples and demonstrate the engineering relevance for practical applications. Results: The newly derived mortar formulation performs well. We tested the new formulation on fully three-dimensional examples of engineering relevance discretized by high-order finite elements up to degrees of p = 10 and found the reproduction of both eigenvalues and eigenmodes to be accurate. Moreover, the mortar method allows for a significant reduction in the algorithmic complexity of mesh generation while simultaneously reducing the overall computational effort. Conclusion: The newly derived modified mortar method for replacing an elastomer layer is not only an academically interesting variant but is capable of solving problems of practical importance in modal-analysis of cross-laminated timber structures.
Computers & Mathematics with Applications, 2015
Contact problems in solid mechanics are traditionally solved using the h-version of the finite el... more Contact problems in solid mechanics are traditionally solved using the h-version of the finite element method. The constraints are enforced along the surfaces of e.g. elastic bodies under consideration. Standard constraint algorithms include penalty methods, Lagrange multiplier methods and combinations thereof. For complex scenarios, a major part of the solution time is taken up by operations to identify points that come into contact. This paper presents a novel approach to model frictionless contact using high order finite elements. Here, we employ an especially designed material model that is inserted into two-respectively threedimensional regions surrounding contacting bodies. Contact constraints are thus enforced on the same manifold as the accompanying structural problem. The application of the current material formulation leads to a regularization of the Karush-Kuhn-Tucker conditions. Our formulation can be classified as a barrier-type method. Results are obtained for two-and three-dimensional problems, including a Hertzian contact problem. Comparisons to a commercial FEA package are provided. The proposed formulation works well for non-matching discretizations on adjacent contact interfaces and handles self-contact naturally. Since the non-penetrating conditions are solved in a physically consistent manner, there is no need for an explicit contact search.
International Journal of Structural Stability and Dynamics, 2015
In this contribution, the finite cell method (FCM) is applied to solve transient problems of line... more In this contribution, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semi-discrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A 1D benchmark problem is solved using FCM, illustrating the method's ability to solve problems of linear elastodynamics obtaining high rates of convergence. Furthermore, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to the results obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics, offering a high degree of accuracy at a moderate com...
Lecture Notes in Computational Science and Engineering
In this contribution the use of hexahedral elements for the structural simulation in a fluid stru... more In this contribution the use of hexahedral elements for the structural simulation in a fluid structure interaction framework is presented, resulting in a consistent kinematic and geometric description of the solid. In order to compensate the additional numerical effort of the three-dimensional approach, an anisotropic p-adaptive method for linear elastodynamic problems is proposed, resulting in a clearly higher efficiency and higher convergence rates than uniform p-extensions. Special emphasis is placed on the accurate transfer of loads considering the fluid discretization for computation of the surface load integrals. For a coupling with a cartesian grid based Lattice Boltzmann code it was found that oscillations in the interface tractions may excite higher structural modes possibly leading to a nonstable coupling behavior. A first remedy to this problem was a linear modal analysis of the structure, thus allowing to control the number of modes to be considered without disregarding bidirectional fluid structure interactions. Preliminary results are presented for the FSI benchmark configuration proposed in this book.
After a short discussion of recent discretization techniques for the lattice-Boltzmann equations ... more After a short discussion of recent discretization techniques for the lattice-Boltzmann equations we motivate and discuss some alternative approaches using implicit, nonuniform FD discretization and mesh refinement techniques. After presenting results of a stability analysis we use an implicit approach to simulate a boundary layer test problem. The numerical results compare well to the reference solution when using strongly refined meshes. Some basic ideas for a nonuniform mesh refinement (with non-cartesian mesh topology) are introduced using the standard discretization procedure of alternating collision and propagation.
International Journal for Numerical Methods in Engineering, 2014
We present a parameter-free domain sewing approach for low-as well as high-order finite elements.... more We present a parameter-free domain sewing approach for low-as well as high-order finite elements. Its final form contains only primal unknowns, i.e., the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem dependent parameters which require an estimation. The presented approach is symmetry-preserving, i.e. the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization and we demonstrate high order accuracy for problems of non-matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p-and B-spline versions of the finite cell method, an embedded domain method of high order, and compared to more classical methods such as the penalty method or Nitsche's method.
The development of flow instabilities due to high Reynolds number flow in artificial heart valve ... more The development of flow instabilities due to high Reynolds number flow in artificial heart valve geometries inducing high strain rates and stresses often leads to hemolysis and related highly undesired effects. Geometric and functional optimization of artificial heart valves is therefore mandatory. In addition to experimental work in this field it is meanwhile possible to obtain increasing insight into flow dynamics by computer simulation of refined model problems. After giving an introductory overview we report the results of the simulation of three-dimensional transient physiological flows in fixed geometries similar to a CarboMedics bileaflet heart valve at different opening angles. The visualization of emerging complicated flow patterns gives detailed information about the transient history of the systems dynamical stability. Stress analysis indicates temporal shear stress peaks even far away from walls. The mathematical approach used is the Lattice Boltzmann method. We obtained reasonable results for velocity and shear stress fields. The code is implemented on parallel hardware in order to decrease computation time. Finally, we discuss problems, shortcomings and possible extensions of our approach.
IOP Conference Series: Materials Science and Engineering, 2010
The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM,... more The Finite Cell Method (FCM), which combines the fictitious domain concept with high-order p-FEM, permits the effective solution of problems with very complex geometry, since it circumvents the computationally expensive mesh generation and guarantees exponential convergence rates for smooth problems. The present contribution deals with the coupling of the FCM approach, which has been applied so far only to linear elasticity, with established nonlinear finite element technology. First, it is shown that the standard p-FEM based FCM converges poorly in a nonlinear formulation, since the presence of discontinuities leads to oscillatory solution fields. It is then demonstrated that the essential ideas of FCM, i.e. exponential convergence at virtually no meshing cost, can be achieved in the geometrically nonlinear setting, if high-order Legendre shape functions are replaced by a hierarchically enriched B-spline patch.
International Journal for Numerical Methods in Engineering, 2009
Over the last decade the Lattice Boltzmann method, which was derived from the kinetic gas theory,... more Over the last decade the Lattice Boltzmann method, which was derived from the kinetic gas theory, has matured as an efficient approach for solving Navier-Stokes equations. The p-FEM approach has proved to be highly efficient for a variety of problems in the field of structural mechanics. Our goal is to investigate the validity and efficiency of coupling the two approaches to simulate transient bidirectional Fluid-Structure interaction problems with geometrically non-linear structural deflections. A benchmark configuration of self-induced large oscillations for a flag attached to a cylinder can be accurately and efficiently reproduced within this setting. We describe in detail the force evaluation techniques, displacement transfers and the algorithm used to couple these completely different solvers as well as the results, and compare them with a benchmark reference solution computed by a monolithic finite element approach.
International Journal for Numerical Methods in Engineering, 2012
The Finite Cell Method (FCM) is a fictitious domain approach based on hierarchical Ansatz spaces ... more The Finite Cell Method (FCM) is a fictitious domain approach based on hierarchical Ansatz spaces of higher order. The method avoids time-consuming and often error-prone mesh-generation and favorably exploits Cartesian grids to embed structures of complex geometry in a simple shaped computational domain thus shifting parts of the computational effort from mesh generation to the computation within the embedding finite cells of regular shape. This paper presents an effective integration approach for voxel-based models of linear elasticity that drastically reduces the computational effort on cell level. The applied strategy allows the pre-computation of an essential part of the cell matrices and vectors of higher order, representing stiffness and load, respectively. Several benchmark problems show the potential of the proposed method in particular for heterogeneous material properties as common in biomedical applications based on computer tomography scans. The applied strategy ensures a fast computation for time-critical simulations and even allows user-interactive simulations for models of moderate size at a high level of accuracy.
A three-dimensional model of the human tibia bone subject to realistic static physiological condi... more A three-dimensional model of the human tibia bone subject to realistic static physiological conditions was developed using the p-version of the ÿnite-element method. The p-version of the ÿnite-element method in combination with the blending-function method enabled the construction of elements displaying accurate representation of the domain's boundary. This allowed the modeling of complex structures like bones using few elements and circumvented the problems of complexity and inaccuracy that are typical of current commercially available ÿnite-element mesh generators. In these methods large p-elements are integrated with high-order functions, permitting the coupling of the mechanical properties of each integration point directly to a voxel matrix extracted from computerized tomographies. This allowed order of 1000 di erent properties to be assigned per element thereby rendering a highly heterogeneous bone model. The ÿnal objective of this bone model is to o ers a platform for the study medical devices and bone composition and behavior in fracture processes and in pathologies as osteoporosis. The results show realistic stresses and displacements of the human tibia under static loading. The stresses are distributed along the whole bone and more pronounced towards the periphery, where there is a larger concentration of cortical bone.
... to give reasonable results, when the solids velocity is small compared to the lattice distanc... more ... to give reasonable results, when the solids velocity is small compared to the lattice distance times ... goal was to test the suitability of LB methods for a coupled fluidstructure flow problem in ... presented here using the LB method gave reasonable results for the prediction of velocity ...
Computers & Mathematics with Applications, 2014
In the case of dominating convection, standard Bubnov-Galerkin finite elements are known to deliv... more In the case of dominating convection, standard Bubnov-Galerkin finite elements are known to deliver oscillating discrete solutions for the convection-diffusion equation. This paper demonstrates that increasing the polynomial degree (p-extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given Péclet number at the nodes of a mesh.
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