adam zdunek - Academia.edu (original) (raw)

Papers by adam zdunek

Research paper thumbnail of To ℘ or not to p – the mixed displacement–pressure p, versus the higher order ℘ displacement finite element formulation, for nearly incompressible linear elasticity

Computers & Mathematics with Applications

Research paper thumbnail of An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Computational Methods in Applied Mathematics

We present an overlapping domain decomposition iterative solver for linear systems resulting from... more We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required ac...

Research paper thumbnail of An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Computational methods in applied mathematics, Jul 12, 2023

We present an overlapping domain decomposition iterative solver for linear systems resulting from... more We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh.

Research paper thumbnail of Acoustic Scattering Problems

Chapman & Hall/CRC applied mathematics and nonlinear science series, Nov 2, 2007

Research paper thumbnail of Perfectly Matched Layer Formulations

Chapman & Hall/CRC applied mathematics and nonlinear science series, Nov 2, 2007

Research paper thumbnail of A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Computers & mathematics with applications, Nov 1, 2021

Abstract Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressib... more Abstract Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficult task in simulations of viscous flows. We illustrate the method with a few steady state laminar solutions.

Research paper thumbnail of Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications

English | 正體中文 | 简体中文 | 全文筆數/總筆數 : 53432/53838 造訪人次 : 14915 線上人數 : 6. RC Version 4.0 © Powered By... more English | 正體中文 | 简体中文 | 全文筆數/總筆數 : 53432/53838 造訪人次 : 14915 線上人數 : 6. RC Version 4.0 © Powered By DSPACE, MIT. Enhanced by NTU Library IR team. 搜尋範圍 全部OA 進階搜尋. ...

Research paper thumbnail of An h-adaptive mortar finite element method for finite deformation contact with higher order p extension

Computers & mathematics with applications, Apr 1, 2017

In this work we present a generalization of the mortar segment-to-segment method for finite defor... more In this work we present a generalization of the mortar segment-to-segment method for finite deformations contact to an h-adaptive version with possible p extension, i.e. using higher order approximation. We recall the main ideas of the mortar algorithm and present the key aspects of adaptivity: error estimation and an h-adaptive strategy. The p extension exploits the feature of the hp-adaptive code in which the contact solver is implemented to handle meshes with nodes of nonuniform orders. We use it to set interior nodes to higher order while leaving linear boundary contact nodes which can be processed by the standard mortar algorithm. Accuracy of elements with low order nodes is restored by adequate subdividing of these elements. Adaptivity and p extension are illustrated with a few numerical tests.

Research paper thumbnail of A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Comput. Math. Appl., 2021

Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscou... more Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficu...

Research paper thumbnail of Two-Grid hp Solver

Research paper thumbnail of On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

Computer Methods in Applied Mechanics and Engineering

Research paper thumbnail of Electromagnetic Scattering Problems

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, 2007

Research paper thumbnail of 3D hp Finite Element Method

Research paper thumbnail of Simulation of Resistivity Logging Devices

Research paper thumbnail of Perfectly Matched Layer Formulations

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of High-Performance Computation

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of Acoustic Scattering Problems

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of A Domain Decomposition Based Parallel Implementation

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of Boundary Value Problems

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, 2007

Research paper thumbnail of Tests with FALKSOL A massively parallel multi-level domain decomposing direct solver

Computers & Mathematics with Applications, 2021

Abstract An implementation of a massively parallel domain decomposing direct finite element equat... more Abstract An implementation of a massively parallel domain decomposing direct finite element equation solver named FALKSOL is tested using the parallel version of the state of the art open-source solver MUMPS (MUltifrontal Massively Parallel Solver) as a reference. FALKSOL includes its own very advanced multi-level domain decomposition and load balancing procedure. In the elemental mode, MUMPS uses a single level domain decomposition, and then solves the interface problem. FALKSOL offers a full out-of-core functionality, while MUMPS offers only a partial one. The problem size that can be solved by MUMPS is therefore hard limited by the amount of physical memory plus swap space 1 present. This is an actual bottleneck. FALKSOL can solve much larger problems. It is essentially only limited by the size of the secondary storage available. With modern so-called Non-Volatile Memory express (NVMe) on the PCI bus, enough I/O bandwidth is obtained with a couple of raided units. The results show that MUMPS is about twice as fast as FALKSOL in the limited range of problem size it copes with. The solvers are complementary. The out-of-core multi-level domain decomposition algorithm in FALKSOL makes it scale. The test indicates that there is a considerable potential gain in scalability/elasticity choosing the FALKSOL multi-level domain-decomposition out-of-core approach. Especially considering more powerful and larger compute systems. The price to be paid in speed compared with MUMPS is reasonable. The current stand-alone implementation of FALKSOL is interfaced with the 3Dhp code Demkowicz (2007) [32] , Demkowicz et al. (2008) [33] . The described MUMPS-like interface can be used to select the best solver for the task.

Research paper thumbnail of To ℘ or not to p – the mixed displacement–pressure p, versus the higher order ℘ displacement finite element formulation, for nearly incompressible linear elasticity

Computers & Mathematics with Applications

Research paper thumbnail of An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Computational Methods in Applied Mathematics

We present an overlapping domain decomposition iterative solver for linear systems resulting from... more We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required ac...

Research paper thumbnail of An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D

Computational methods in applied mathematics, Jul 12, 2023

We present an overlapping domain decomposition iterative solver for linear systems resulting from... more We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh.

Research paper thumbnail of Acoustic Scattering Problems

Chapman & Hall/CRC applied mathematics and nonlinear science series, Nov 2, 2007

Research paper thumbnail of Perfectly Matched Layer Formulations

Chapman & Hall/CRC applied mathematics and nonlinear science series, Nov 2, 2007

Research paper thumbnail of A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Computers & mathematics with applications, Nov 1, 2021

Abstract Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressib... more Abstract Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficult task in simulations of viscous flows. We illustrate the method with a few steady state laminar solutions.

Research paper thumbnail of Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume II Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications

English | 正體中文 | 简体中文 | 全文筆數/總筆數 : 53432/53838 造訪人次 : 14915 線上人數 : 6. RC Version 4.0 © Powered By... more English | 正體中文 | 简体中文 | 全文筆數/總筆數 : 53432/53838 造訪人次 : 14915 線上人數 : 6. RC Version 4.0 © Powered By DSPACE, MIT. Enhanced by NTU Library IR team. 搜尋範圍 全部OA 進階搜尋. ...

Research paper thumbnail of An h-adaptive mortar finite element method for finite deformation contact with higher order p extension

Computers & mathematics with applications, Apr 1, 2017

In this work we present a generalization of the mortar segment-to-segment method for finite defor... more In this work we present a generalization of the mortar segment-to-segment method for finite deformations contact to an h-adaptive version with possible p extension, i.e. using higher order approximation. We recall the main ideas of the mortar algorithm and present the key aspects of adaptivity: error estimation and an h-adaptive strategy. The p extension exploits the feature of the hp-adaptive code in which the contact solver is implemented to handle meshes with nodes of nonuniform orders. We use it to set interior nodes to higher order while leaving linear boundary contact nodes which can be processed by the standard mortar algorithm. Accuracy of elements with low order nodes is restored by adequate subdividing of these elements. Adaptivity and p extension are illustrated with a few numerical tests.

Research paper thumbnail of A discontinuous Petrov-Galerkin method for compressible Navier-Stokes equations in three dimensions

Comput. Math. Appl., 2021

Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscou... more Application of a Discontinuous Petrov-Galerkin (DPG) method for simulation of compressible viscous flows in three dimensions is presented. The approach enables construction of stable schemes for problems with a small perturbation parameter. The main idea of the method is a weak formulation with a relaxed interelement continuity of the solution. The formulation satisfies the inf-sup condition with the stability constant independent of the small perturbation parameter, which here is the viscosity constant for the compressible Navier-Stokes equations. The DPG discrete formulation uses the specially designed so-called optimal test functions. They do not compromise the inf-sup stability of the continuous formulation. DPG does not use any artificial dissipation for the compressible Navier-Stokes equations. Being a residual minimization method it has got a built-in a posteriori error estimation which allows for mesh adaptivity leading to resolving reliable viscous fluxes, the major difficu...

Research paper thumbnail of Two-Grid hp Solver

Research paper thumbnail of On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

Computer Methods in Applied Mechanics and Engineering

Research paper thumbnail of Electromagnetic Scattering Problems

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, 2007

Research paper thumbnail of 3D hp Finite Element Method

Research paper thumbnail of Simulation of Resistivity Logging Devices

Research paper thumbnail of Perfectly Matched Layer Formulations

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of High-Performance Computation

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of Acoustic Scattering Problems

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of A Domain Decomposition Based Parallel Implementation

Computing with hp-ADAPTIVE FINITE ELEMENTS, 2007

Research paper thumbnail of Boundary Value Problems

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, 2007

Research paper thumbnail of Tests with FALKSOL A massively parallel multi-level domain decomposing direct solver

Computers & Mathematics with Applications, 2021

Abstract An implementation of a massively parallel domain decomposing direct finite element equat... more Abstract An implementation of a massively parallel domain decomposing direct finite element equation solver named FALKSOL is tested using the parallel version of the state of the art open-source solver MUMPS (MUltifrontal Massively Parallel Solver) as a reference. FALKSOL includes its own very advanced multi-level domain decomposition and load balancing procedure. In the elemental mode, MUMPS uses a single level domain decomposition, and then solves the interface problem. FALKSOL offers a full out-of-core functionality, while MUMPS offers only a partial one. The problem size that can be solved by MUMPS is therefore hard limited by the amount of physical memory plus swap space 1 present. This is an actual bottleneck. FALKSOL can solve much larger problems. It is essentially only limited by the size of the secondary storage available. With modern so-called Non-Volatile Memory express (NVMe) on the PCI bus, enough I/O bandwidth is obtained with a couple of raided units. The results show that MUMPS is about twice as fast as FALKSOL in the limited range of problem size it copes with. The solvers are complementary. The out-of-core multi-level domain decomposition algorithm in FALKSOL makes it scale. The test indicates that there is a considerable potential gain in scalability/elasticity choosing the FALKSOL multi-level domain-decomposition out-of-core approach. Especially considering more powerful and larger compute systems. The price to be paid in speed compared with MUMPS is reasonable. The current stand-alone implementation of FALKSOL is interfaced with the 3Dhp code Demkowicz (2007) [32] , Demkowicz et al. (2008) [33] . The described MUMPS-like interface can be used to select the best solver for the task.