1st Edition

Advanced Numerical Simulation Methods
From CAD Data Directly to Simulation Results

ISBN 9781138026346
Published August 17, 2015 by CRC Press
326 Pages

USD $120.00

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Book Description

This book is an entertaining, easy to read introduction to advanced numerical modeling. The aim of the book is to lead the reader on a journey towards the ‘holy grail’ of numerical simulation, namely one without the requirement of mesh generation, that takes data directly from CAD programs. On this journey readers will discover the beauty of Non-uniform Rational B-Splines or NURBS and see how well they are suited for describing geometry, discover how CAD programs work and how their data can be used for simulation. The implementation of NURBS-based simulation is discussed using Finite Element and Boundary Element technology. This is a hands-on book with emphasis on implementation and examples of programming in a higher level language are given. It has been written for users of simulation software, so they can understand the benefits of this new technology and demand progress from a somewhat conservative industry, for software developers, so they can see that this is a technology with a big future and for researchers, in the hope that it will attract more people to work in this exciting new field.

Table of Contents

1. Introduction                                                                                  
1. A brief history of simulation                                                                 
1.1. The world’s first simulation                                                               
1.2. Emergence of mathematics and mechanics                                            
1.3. Computer age                                                                                
2. Basic steps in simulation                                                                     
2.1. Geometry description                                                                        
2.2. Approximation of the unknown                                                      
2.3. Solution                                                                                    
2.4. Recovery of the results                                                                    
3. A change of paradigm: towards a more efficient and accurate simulation                                                                                
4. Organization of the text
2. Stage 1: Basis functions                                                                      
1. One-dimensional basis functions                                                              
1.1. Lagrange and Serendipity functions                                                     
1.2. From B-splines to NURBS                                                                   
2. Two-dimensional basis functions                                                              
2.1. Lagrange and Serendipity functions                                                     
2.2. B-splines                                                                                  
2.3. NURBS                                                                                      
2.4. T-splines                                                                                  
3. Programming                                                                                   
4. The NURBS toolkit                                                                             
5. Summary and conclusions
3. Stage 2: Geometry                                                                             
1. Coordinate systems                                                                            
1.1. Coordinate transformation                                                               
2. Curves                                                                                        
2.1. Mapping with Serendipity/Lagrange basis functions                           
2.2. Mapping with NURBS                                                                    
3. Programming                                                                                   
3.1. NURBS toolkit                                                                               
3.2. Geometry functions                                                                         
3.3. Examples                                                                                    
3.4. Example 1: Circular arc                                                                   
3.5. Example 2: Horseshoe tunnel                                                          
3.6. Example 3: Plate with hole                                                                 
4. Surfaces                                                                                      
4.1. Mapping with Serendipity/Lagrange basis functions                           
4.2. Mapping with NURBS basis functions                                                
4.3. Programming                                                                                 
5. Surface of revolution                                                                         
5.1. Example 1: Cylindrical surface                                                            
5.2. Example 2: Spherical surface                                                               
5.3. Example 3: Bell shaped surface                                                           
6. Lofted surfaces                                                                               
7. NURBS surfaces with cutouts                                                                   
7.1. Analysis suitable trimmed NURBS surfaces                                        
8. Infinite NURBS patch                                                                          
8.1. Example                                                                                     
9. Summary and conclusions 
4. Stage 3: Computer Aided Design                                                                
1. Introduction                                                                                  
2. IGES data structure                                                                           
3. How CAD programs describe geometry – entity types                               
3.1. Line entity (110)                                                                          
3.2. Surface of revolution entity (120)                                                      
3.3. Rational B-spline entity (126)                                                       
3.4. Rational B-spline surface entity (128)                                           
3.5. Boundary entity (141)                                                                   
4. NURBS surfaces                                                                                
5. Trimmed NURBS surfaces                                                                        
6. Summary and conclusions
5. Stage 4: Introduction to numerical simulation                                          
1. One-dimensional simulation                                                                   
1.1. Ritz method                                                                                 
1.2. Approximation                                                                               
2. Steps in the simulation                                                                       
2.1. Description of the geometry                                                          
2.2. Description of known values                                                              
2.3. Convergence tests                                                                          
2.4. Approximation of unknown                                                          
2.5. P-refinement or order elevation                                                          
2.6. H-refinement, the Finite Element Method                                         
2.7. Knot insertion, isogeometric method                                                 
2.8. K-refinement                                                                               
2.9. Summary and conclusions                                                             
3. 2-D simulation, plane stress and plane strain                                          
3.1. Boundary Conditions (BC)                                                                 
3.2. Using one NURBS patch                                                                    
3.3. Comparison with classical FEM                                                         
3.4. Example                                                                                     
3.5. Multiple NURBS patches                                                                   
3.6. Bezièr elements                                                                            
3.7. Trimmed NURBS patches                                                                  
3.8. Convergence test                                                                           
4. Summary and conclusions
6. Stage 5: Plates and shells                                                                    
1. Kirchhoff plate                                                                               
1.1. Plates                                                                                     
1.2. Examples                                                                                    
2. Kirchhoff shells                                                                              
2.1. Example 1: Scordelis roof                                                                  
2.2. Example 2: Trimmed Scordelis roof                                                  
2.3. Example 3: Arched Scordelis roof                                                     
3. Multiple patches                                                                              
3.1. Assembly                                                                                   
3.2. Example                                                                                     
4. Summary and conclusions
7. Stage 6: Integral equations                                                                   
1. Introduction                                                                                  
2. Trefftz method                                                                                
2.1. Example                                                                                     
2.2. Conclusions                                                                                 
3. Integral equations                                                                            
3.1. Theorem of Betti                                                                           
3.2. Rigid body trick                                                                            
3.3. Conclusions                                                                                 
4. Numerical solution of integral equations                                                
4.1. Nyström method                                                                            
4.2. Galerkin method                                                                            
4.3. Collocation                                                                                 
4.4. Discretisation                                                                              
5. Summary and conclusions 
8. Stage 7: The boundary element method for plane problems                  
1. Introduction                                                                                  
2.  Classical isoparametric approach                                                            
2.1. Numerical evaluation of integrals                                                 
3. NURBS based approach                                                                          
3.1. Boundary conditions                                                                      
4. Assembly of multiple patches                                                                  
4.1. Pure Neumann problem                                                                 
4.2. Mixed Neumann/Dirichlet problem                                              
4.3. Symmetry                                                                                   
5. Postprocessing                                                                                
5.1. Results on the boundary                                                                
5.2. Results inside the domain                                                              
6. Programming                                                                                   
7. Examples                                                                                      
7.1. Potential problem: Flow past isolator                                            
7.2. Elasticity problem: Circular excavation in infinite domain               
7.3. Practical example: Horseshoe tunnel                                             
8. Conclusions     
9. Stage 8: The boundary element method for three-dimensional problems                                                               
1. Introduction                                                                                  
2. Numerical integration                                                                         
2.1. Regular integration                                                                        
2.2. Determination of the optimal number of Gauss points                  
2.3. Regular integration                                                                        
2.4. Nearly singular integration                                                            
2.5. Weakly singular integration                                                           
2.6. Infinite patches                                                                            
3. Symmetry                                                                                      
4. Multiple patches                                                                              
5. Postprocessing                                                                                
5.1. Stress recovery                                                                            
5.2. Internal stress computation                                                            
6. Test examples                                                                                 
6.1. Infinite tunnel                                                                             
6.2. Loading on infinite half-space                                                       
7. Examples                                                                                      
7.1. Infinite tunnel in infinite domain near tunnel face                            
7.2. Finite tunnel in a semi-infinite domain                                              
7.3. Branched tunnel                                                                             
8. Conclusions    
10 Stage 9: The boundary element method with volume effects                 
1. Introduction                                                                                  
2. Effect of body forces and initial strain                                                      
2.1. Body forces                                                                                
2.2. Effect of initial strain                                                                   
2.3. Solution                                                                                    
3. Implementation for plane problems                                                        
3.1. Geometry definition                                                                      
3.2. Computation of the volume integral                                              
4. Implementation for 3-D problems                                                           
4.1. Geometry definition                                                                      
4.2. Computation of the volume integral                                              
5. Iterative solution algorithm                                                                  
6. Inclusions                                                                                    
6.1. Computation of body force                                                            
6.2. Steps in the analysis                                                                      
7. Inelastic behavior                                                                            
7.1. Yield conditions                                                                           
7.2. Determination of plastic strain increment                                     
8. Implementation of plasticity for plane problems                                     
8.1. Determination of plastic zone                                                             
8.2. Computation of the volume term                                                   
8.3. Numerical integration                                                                    
8.4. Internal stress computation                                                            
8.5. Extension of plastic zone during iteration                                          
9. Implementation for 3-D problems                                                           
9.1 Determination of the plastic zone                                                       
9.2 Computation of the volume term                                                   
9.3 Numerical integration                                                                    
10  Programming                                                                                    
11. Example                                                                                        
12. Summary and conclusions
11. Stage 10: The time domain                                                                      
1. Introduction                                                                                  
1.1. Bernoulli beam with mass                                                                  
2. Solutions in the frequency domain                                                             
2.1.  Numerical solution                                                                        
3. Solutions in the time domain                                                                  
3.1. Finite difference method                                                                   
3.2. Newmark method                                                                          
4. Programming                                                                                   
5. Summary and conclusions   
Appendix: Fundamental solutions                                                              
1. Stress solution I:(x,y)                                                                       
2. Derived solution for displacement S(x,y)                                                     
3. Derived solution for traction R(x,y)                                                        
4. Derived solution for displacement S(x,y)                                                     
5. Derived solution for traction R(x,y)                                                        
6. Derivatives of kernel S(x,y)                                                                  
7. Derivatives of kernel R(x,y)

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Gernot Beer has been working in the area of numerical simulation since 1975 and has published 4 books on this topic. He has been involved in applying numerical simulation in numerous projects around the world, including the design of caverns for the CERN facility in Geneva and the investigation of a large underground power station in Iran. After working in a consulting office and for a large scientific and industrial research organisation he was appointed in 1993 as professor for structural analysis at the University of Technology Graz, Austria. Since 2012 he is emeritus professor of this institution.


The book […] comes with a developing toolkit that allows the reader to gain some hands-on experience and to get familiar with the new concepts [….] This book is pioneer on the integration of CAD and Boundary Element Analysis.

Prof. Dr. Adrián P. Cisilino, UNMdP, Mar del Plata, Argentina