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2019 | Book

Introduction Read chapter Read first chapter

Error Estimates for Advanced Galerkin Methods

Author: Dr. Marcus Olavi Rüter

Publisher: Springer International Publishing

Book Series : Lecture Notes in Applied and Computational Mechanics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This monograph provides a compendium of established and novel error estimation procedures applied in the field of Computational Mechanics. It also includes detailed derivations of these procedures to offer insights into the concepts used to control the errors obtained from employing Galerkin methods in finite and linearized hyperelasticity. The Galerkin methods introduced are considered advanced methods because they remedy certain shortcomings of the well-established finite element method, which is the archetypal Galerkin (mesh-based) method. In particular, this monograph focuses on the systematical derivation of the shape functions used to construct both Galerkin mesh-based and meshfree methods. The mesh-based methods considered are the (conventional) displacement-based, (dual-)mixed, smoothed, and extended finite element methods. In addition, it introduces the element-free Galerkin and reproducing kernel particle methods as representatives of a class of Galerkin meshfree methods. Including illustrative numerical examples relevant to engineering with an emphasis on elastic fracture mechanics problems, this monograph is intended for students, researchers, and practitioners aiming to increase the reliability of their numerical simulations and wanting to better grasp the concepts of Galerkin methods and associated error estimation procedures.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In this introductory chapter, the topics of this monograph are briefly discussed and embedded into the bigger picture of computational validation and verification strategies in Computational Mechanics. More precisely, different types of errors are introduced that appear during the numerical simulation process of a physical phenomenon based on various Galerkin methods. The Galerkin methods dealt with in this monograph are the (conventional) finite element method (FEM) and, in particular, advanced versions of the finite element method, such as the extended finite element method (XFEM).
Marcus Olavi Rüter
Chapter 2. Newtonian and Eshelbian Mechanics
Abstract
The objective of this chapter is to present an introduction to the theory of continuum mechanics of elastic structures. Classical continuum mechanics deals with finding the spatial configuration of an elastic body that is subjected to external forces. This forward problem is attributed to Sir Isaac Newton and therefore termed Newtonian mechanics. In the associated inverse problem, which is attributed to John Douglas Eshelby and therefore termed Eshelbian mechanics, we are concerned with the forces applied to the spatial configuration. In Newtonian mechanics, the applied forces are of a physical nature, and as a result, the associated stress that arises in the spatial configuration of the elastic body is well known as the (physical) Cauchy stress. In Eshelbian mechanics, on the other hand, the deformed elastic body is subjected to so-called material forces, and the resulting stress in the initial configuration (of the forward problem) is termed the (material) Eshelby stress. In this chapter, the stress tensors that naturally appear in both Newtonian and Eshelbian mechanics are systematically derived for both compressible and (nearly) incompressible materials.
Marcus Olavi Rüter
Chapter 3. Boundary Value Problems
Abstract
In this chapter, the field equations derived in the previous chapter are summarized and supplemented by boundary conditions. This results in the boundary value problems of compressible and (nearly) incompressible finite hyperelasticity within both Newtonian and Eshelbian mechanics. The derivations are performed in terms of their strong and weak forms and supplemented by appropriate linearizations that are used within the iterative Newton-Raphson scheme. In the special case of the first iteration, this yields the linearized elasticity problem. It will also be demonstrated how further simplifications result in the Poisson (membrane) and uniaxial problems. Attention is focused in this chapter on the derivations of the variational forms of the boundary value problems. These forms are also known as the weak forms because the strong forms only need to be satisfied in an integral rather than in a pointwise sense. Moreover, integration by parts reduces the differentiability requirements for the functions involved in the weak forms. These are the key ingredients needed to develop numerical methods of the Galerkin type, as will be presented in the subsequent chapter.
Marcus Olavi Rüter
Chapter 4. Galerkin Methods
Abstract
Having derived in the previous chapter various boundary value problems, including the finite and linearized hyperelasticity problems for both compressible and (nearly) incompressible materials, a reasonable question is how these problems can be solved. For most cases in engineering practice, the problems, including their geometry, are too complex for the feasible derivation of an exact analytical solution even though such a solution exists. We are therefore forced to employ numerical methods to obtain, at least, approximate solutions to the boundary value problems stated in the previous chapter.
Marcus Olavi Rüter
Chapter 5. Numerical Integration
Abstract
This chapter provides a brief account of numerical integration schemes used to approximately evaluate definite integrals of arbitrary functions. Numerical integration schemes are required to evaluate the integrals that appear in the Galerkin weak forms presented in the preceding chapter for both mesh-based and meshfree methods. First, the classical Gauss quadrature scheme is explained before the more modern stabilized conforming nodal integration (SCNI) scheme is derived. Stabilized conforming nodal integration is a more advanced domain integration scheme that relies on a modification of the Galerkin weak form. A firm theoretical foundation for the modification of the Galerkin weak form used in stabilized conforming nodal integration is provided by the enhanced assumed strain (EAS) method. This method was originally introduced to alleviate volumetric locking in the finite element method. In this chapter, it is demonstrated how the enhanced assumed strain method can be used as a basis for a nodal integration scheme that can be applied to both Galerkin mesh-based and meshfree methods.
Marcus Olavi Rüter
Chapter 6. Energy Norm A Posteriori Error Estimates
Abstract
This chapter provides deeper insights into the verification part of the computational V&V strategy introduced in Sect. 1.1. To be more precise, we investigate the question whether the boundary value problems derived in Chaps. 2 and 3 are solved right by the Galerkin methods presented in Chaps. 4 and 5, i.e. the (mixed) finite element method (based on SCNI), the extended finite element method, and the meshfree element-free Galerkin and reproducing kernel particle methods. For the time being, we restrict our considerations to the linearized elasticity problem (3.28) because this linear problem allows for the development of verification strategies in a more convenient way. Verification strategies applied to the finite hyperelasticity problem will be detailed in Chap. 8.
Marcus Olavi Rüter
Chapter 7. Goal-oriented A Posteriori Error Estimates in Linearized Elasticity
Abstract
In this chapter, attention is focused on goal-oriented error estimation procedures that are based on generalized error measures because they are, in most cases, of greater interest to engineers than their energy norm counterparts. The error estimation procedures presented in the previous chapter for both Galerkin mesh-based and meshfree methods are extended in this chapter to provide an estimation of the generalized error measure. This error measure is typically related to the error of a given quantity of interest that naturally occurs in the design-specific computation of an engineering model within the computational V&V strategy. Examples of quantities of interest are local displacement or stress distributions and the fracture criterion in fracture mechanics problems, which is frequently a nonlinear quantity of interest that requires a linearization. To estimate the error of a given quantity of interest, a so-called dual problem needs to be solved in addition to the primal problem, which is, in this chapter, the conventional linearized elasticity problem. Moreover, a multi-space strategy is introduced that can use used to solve the dual problem on a different mesh or particle distribution and adds versatility to the goal-oriented error estimation procedures.
Marcus Olavi Rüter
Chapter 8. Goal-oriented A Posteriori Error Estimates in Finite Hyperelasticity
Abstract
Coming full circle in this chapter, expansions of the goal-oriented error estimation procedures presented in the preceding chapter to the finite hyperelasticity problem within both Newtonian and Eshelbian mechanics are derived for compressible and (nearly) incompressible materials. These error estimation procedures represent the most challenging ones presented in this monograph from both theoretical and numerical points of view. As a consequence, attention is focused on the derivation of error approximations rather than upper- or lower-bound error estimates. In the nonlinear case, a natural norm, such as the energy norm does not exist. The estimation of the general error measures introduced in the preceding chapter, on the other hand, does not necessarily rely on norm-based error estimators and thus allows for the derivation of a more versatile approach in a posteriori error estimation that can be employed in this chapter. Throughout this chapter, we confine ourselves to Galerkin mesh-based methods although similar error estimation procedures can also be developed for Galerkin meshfree methods.
Marcus Olavi Rüter
Chapter 9. Numerical Examples
Abstract
In this chapter, various numerical examples are presented that demonstrate the numerical performance of the a posteriori error estimators developed in this monograph for both the finite and linearized hyperelasticity problems and the Poisson problem. For the energy norm and related error estimators, examples with different types of singularities are considered. The goal-oriented error estimators are primarily applied to linear and nonlinear elastic fracture mechanics problems, including crack propagation, because the J-integral, as a fracture criterion, serves as a numerically challenging nonlinear quantity of particular engineering interest. The numerical methods considered in this chapter are the conventional, mixed, dual-mixed, and extended finite element methods, the finite element method based on stabilized conforming nodal integration (SCNI), and the element-free Galerkin and reproducing kernel particle methods. Various materials, such as concrete and aluminum, are investigated in this chapter with an emphasis on glass and rubber. Although these materials seem to exhibit different material behavior, they share many similarities.
Marcus Olavi Rüter
Backmatter
Metadata
Title
Error Estimates for Advanced Galerkin Methods
Author
Dr. Marcus Olavi Rüter
Copyright Year
2019
Electronic ISBN
978-3-030-06173-9
Print ISBN
978-3-030-06172-2
DOI
https://doi.org/10.1007/978-3-030-06173-9

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