Verifying Calculations - Forty Years On: An Overview Of Classical Verification Techniques For Fem Simulations (springerbriefs In Applied Sciences And Technology)

This work provides an overview of __a posteriori__ error assessment techniques for Finite Element (FE) based numerical models. These tools aim at estimating and controlling the discretization error in scientific computational models, being the basis for the numerical verification of the FE solutions. The text discusses the capabilities and limitations of classical methods to build error estimates which can be used to control the quality of numerical simulations and drive adaptive algorithms, with a focus on Computational Mechanics engineering applications. Fundamentals principles of residual methods, smoothing (recovery) methods, and constitutive relation error (duality based) methods are thus addressed along the manuscript. Attention is paid to recent advances and forthcoming research challenges on related topics. The book constitutes a useful guide for students, researchers, or engineers wishing to acquire insights into state-of-the-art techniques for numerical verification. Preface 6 Contents 8 Explicit Residual Methods 9 1 Introduction 9 2 Fourier Approximation 11 2.1 A Simple Elliptic Problem 12 2.2 A Posteriori Analysis in the Periodic Setting 13 2.3 Stopping Criteria for Iterative Solution Algorithms 13 3 Finite Element Approximation 14 3.1 Evaluation of the Residual 16 3.2 Extension to the Stokes Problem 18 4 Reduced Basis Approximation 19 4.1 Basics and Rational of the Reduced Basis Approach 19 4.2 A Posteriori Validation of Outputs 21 4.3 Black Box Implementation 24 References 25 Implicit Residual Type Error Estimators 27 1 Introduction and Problem Statement 27 1.1 Preliminaries 27 1.2 Problem Statement 28 2 Error Equations and a Priori Estimates 30 3 Implicit Residual Error Estimates 32 3.1 Classification 32 3.2 Assessing the Energy Norm of the Error 34 3.3 Element Residual Method; Equilibrated Residual Estimates 35 3.4 Subdomain Residual Methods; Flux-Free Estimates 36 4 Goal-Oriented Estimates 37 4.1 Lower Bounds for the Energy Using Implicit Dirichlet Estimates 39 References 40 Fundaments of Recovery-Based Error Estimation and Bounding 41 1 Introduction 41 2 Recovery-Based Error Estimators 43 2.1 The SPR Technique 43 2.2 A Nearly Equilibrated Recovery Procedure. The SPR-C Technique 45 2.3 Nearly Equilibrated Displacement Recovery Procedure. The SPR-CD Technique 47 2.4 Singular Fields 50 3 Error Estimation and Bounding 52 3.1 Error Estimation in Energy Norm 52 3.2 Error Estimation in Quantities of Interest 52 3.3 A SPR-Based Upper Bounding Technique 53 3.4 Error Lower Bounds in Energy Norm 57 4 Error Estimation in the Recovered Solution Field 58 4.1 Error Norm Representation for the Recovered Solution 59 4.2 h-adaptive Refinement Process 61 5 Numerical Results 62 6 Conclusions 64 References 64 The Constitutive Relation Error Method: A General Verification Tool 66 1 Introduction 66 2 Reference Problem and Notations 68 3 Use of CRE for Elasticity Problems 69 3.1 Basics on CRE and Global Error Estimation 69 3.2 Goal-Oriented Error Estimation 71 4 Use of CRE for Visco-Plasticity Problems 73 4.1 Basics on CRE and Global Error Estimation 73 4.2 Goal-Oriented Error Estimation 76 5 Getting Accurate and Practical Error Bounds on Outputs of Interest 78 5.1 Non-intrusive Approach for the Adjoint Solution 78 5.2 Application to Pointwise Quantities 80 6 Control of Various Computational Approaches 82 6.1 Stochastic Approach 82 6.2 PGD-Reduced Models 89 7 Conclusions and Prospects 96 References 98 Front Matter....Pages i-vii Explicit Residual Methods....Pages 1-18 Implicit Residual Type Error Estimators....Pages 19-32 Fundaments of Recovery-Based Error Estimation and Bounding....Pages 33-57 The Constitutive Relation Error Method: A General Verification Tool....Pages 59-94