Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

Zahur Ullah

Research output: Other contribution

Abstract

A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions.Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results.
LanguageEnglish
TypePhD Thesis
Number of pages222
Publication statusE-pub ahead of print - 22 Apr 2013

Fingerprint

Galerkin methods
Mechanics
Error analysis
Geomechanics
Recovery
Computer architecture
Biomechanics
Metal forming
Message passing
Constitutive models
Linear equations
Software packages
Plasticity
Entropy
Boundary conditions
Decomposition
Finite element method
Data storage equipment
Communication
Costs

Keywords

  • meshless methods
  • meshfree methods
  • element-free Galerkin method (EFGM)
  • maximum entropy shape functions
  • Moving least squares
  • material nonlinearities
  • geometrical nonlinearities
  • elasto-plasticity
  • finite deformation
  • error estimation
  • adaptivity
  • FE-EFGM coupling
  • superconvergent patch recovery method
  • adaptive FE-EFGM coupling
  • distributed memory parallel programming
  • METIS
  • MUMPS
  • MPI

Cite this

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title = "Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method",
abstract = "A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions.Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results.",
keywords = "meshless methods, meshfree methods, element-free Galerkin method (EFGM), maximum entropy shape functions, Moving least squares, material nonlinearities, geometrical nonlinearities, elasto-plasticity, finite deformation, error estimation, adaptivity, FE-EFGM coupling, superconvergent patch recovery method, adaptive FE-EFGM coupling, distributed memory parallel programming, METIS, MUMPS, MPI",
author = "Zahur Ullah",
year = "2013",
month = "4",
day = "22",
language = "English",
type = "Other",

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Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method. / Ullah, Zahur.

222 p. 2013, PhD Thesis.

Research output: Other contribution

TY - GEN

T1 - Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

AU - Ullah, Zahur

PY - 2013/4/22

Y1 - 2013/4/22

N2 - A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions.Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results.

AB - A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions.Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results.

KW - meshless methods

KW - meshfree methods

KW - element-free Galerkin method (EFGM)

KW - maximum entropy shape functions

KW - Moving least squares

KW - material nonlinearities

KW - geometrical nonlinearities

KW - elasto-plasticity

KW - finite deformation

KW - error estimation

KW - adaptivity

KW - FE-EFGM coupling

KW - superconvergent patch recovery method

KW - adaptive FE-EFGM coupling

KW - distributed memory parallel programming

KW - METIS

KW - MUMPS

KW - MPI

M3 - Other contribution

ER -