Development and Application of the Finite Element Method based on MatLab (eBook)

Herbert Baaser (Autor)

eBook Download: PDF

2010 | 2010
X, 64 Seiten
Springer Berlin (Verlag)
978-3-642-13153-0 (ISBN)

Infos

The intention of this booklet is a brief but general introduction into the treatment of the Finite Element Method (FEM). The FEM has become the leading method in computer-oriented mechanics, so that many scienti?c brancheshavegrownup besides overthelastdecades. Nevertheless,theFEM today is a question of economy. On the one hand its industrial application is forced to reduce product development costs and time, on the other hand a large number of commercial FEM codes and a still growing number of software for e?ective pre- and postprocessors are available in the meantime. Due to that, today it is a quite challenging task to operate with all these di?erent tools at the same time and to understand all handling and so- tion techniques developed over the last years. So, we want to help in getting a deeper insight into the main 'interfaces' between the 'customers of the FEM' and the codes itself by providing a totally open structured FE-code based on Matlab, which is a very powerful tool in operating with matrix based formulations. That idea and conditions forced us some years ago to initiateDAEdalon as a tool for general FE developments in research appli- tions. In spite of still existing high sophisticated - mostly commercial - FE codes, the success and the acceptance of such a structured tool justify that decision afterwards more and more.

Inhalt

Title Page 2
Preface 6
Contents 8
A Quick Start into DAEdalon 10
Download and Installation 10
Elastomechanical Example 10
Principle of Virtual Displacements 11
Discretization of Basic Equations 12
Run DAEdalon 14
Fundamentals of Solid (Continuum) Mechanics 19
Vector–, Matrix– and Tensor–Notation 19
Kinematics and Deformation Gradient F 20
Definitions 20
Properties and Derivatives of F 21
Implementation 23
Strain– and Stress–Tensors 23
A Selection 23
Invariants and Derivatives of Strain Tensors 24
Stress Representation — Voigt Notation 24
Rate of Strain Energy — Stress Power, Internal Energy Turn Over 26
Variational Principle and Weak Form 26
Discretization in Space 27
Preparation and Rearrangement of Equations 27
Linear Shape Functions 28
Derivatives of the Shape Functions 29
Volume Integration 31
General Treatment of Nonlinearities — Solution by Newton–Procedure 33
Linearization 34
Iteration 34
Constitutive Behavior 35
The Materials Point of View 35
Stress Response 35
The Material Modulus ID 35
Selected Constitutive Models 36
Hyperelasticity 36
Parameter Calibration 39
Inelastic Behavior 40
FEM Implementation within {/sc Matlab} 45
{/sc Matlab} Basics 45
Structure of DAEdalon 46
Preprocessing 47
Structure of Input-Files and Processing 47
Numerical Solution 47
Assembly Procedure — The Global System 47
{/sc Newton}–Iteration 48
Postprocessing 49
Mesh Representation, Loads and Boundary Conditions 49
Contour-Plots 49
GUI — Realization of a Graphical Environment 49
Writing Extensions for DAEdalon 50
Continuum Element elem4.m with 4–Nodes in Plane Strain 50
Formulation for Rotational Symmetry of elem4.m 51
EAS Expansion of elem4.m 52
General Truss Element elem10.m with 2 Nodes 55
Material Models 56
Applications and Examples 58
Crane Modeled by 2D–Trusses 58
Axisymmetric Applications 60
Crack Tip Simulation 61
K_I–Field 61
Analysis of Plastic Zone within K_I–Field 64
What Does DAEdalon Mean ?— Background for Computational System — and Greek Mythology 65
References 67
Index 69

Textauszug

"3 Constitutive Behavior (p. 27-28)

3.1 The Materials Point of View

The description of any material behavior within a ?nite element simulation requires a clearly structured interface within an element formulation. As stated before in (2.36), in Sec. 2.6.4 and in (2.53), the (local) material behavior is respected and needed for at the integration points while the integration loop on element level. According to the presented modular structure of DAEdalon, we intend to de?ne any constitutive model by the function MATMOD, where the deformation gradient F and the history database is given as input and the material response is given out in terms of the stress tensor and the material (tangent) modulus.

In that sense, linear or nonlinear material behavior is treated in the same way. So, the user has all possibilities in de?ning any deformation measure as function of the deformation gradient F and the history and take care for the conjugated stress tensor and its derivative. Please note, that the overall convergence behavior of a ?nite element solution procedure depends dramatically on the right formulation of the stress response and its derivative in form of the material modulus.

3.1.1 Stress Response

Basically, one has the free choice in formulating the material behavior with respect to any con?guration. There are many discussions on that topic, each with advances and each with negative aspects, but the overlaying element formulation has to be respected to take care on the energy turn over expressed by the tensors given by the material model, see Sec. 2.4.

3.1.2 The Material Modulus

The crucial task in the formulation and assembly of Kmat in (2.53) is the representation of the derivative of the stress response with respect to the applied deformation measure in the algorithmic setting. We illustrate here the results of that procedure for two typical material behaviors, namely hyperelasticity in its simplest form and ?nite plasticity, in the following. Generally, we have a look on the structure and the implementation of typical representations of forth order tensors like."

Produkt-Details

• Erscheint lt. Verlag: 10.5.2010
• Zusatzinfo: X, 64 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Informatik ► Theorie / Studium ► Künstliche Intelligenz / Robotik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: DAEDalon • Development • Elastodynamics • Elastostatics • Finite Element Method • MATLAB • Software
• ISBN-10: 3-642-13153-0 / 3642131530
• ISBN-13: 978-3-642-13153-0 / 9783642131530