Jože Korelc

Automatic generation of finite-element code by simultaneous optimization of expressions

Abstract The paper presents a MATHEMATICA package SMS (Symbolic Mechanics System) for the automatic derivation of formulas needed in nonlinear finite element analysis. Symbolic generation of the characteristic arrays of nonlinear finite elements (e.g. nodal force vectors, stiffness matrices, sensitivity vectors) leads to exponential behavior, both in time and space. A new approach, implemented in SMS, avoids this problem by combining several techniques: symbolic capabilities of Mathematica, automatic differentiation technique, simultaneous optimization of expressions and a stochastic evaluation of the formulas instead of a conventional pattern matching technique. SMS translates the derived symbolic formulas into an efficient compiled language (FORTRAN or C). The generated code is then incorporated into an existing finite element analysis environment. SMS was already used to developed several new, geometrically and materially nonlinear finite elements with up to 72 degrees of freedom. The design and implementation of SMS are presented. Efficiency of the new approach is compared with the efficiency of the manually written code on an example.

Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation

We first review some recent and current research works attributing to a very significant progress on shell problem theoretical foundation and numerical implementation attained over a period of the last several years. We then discuss theoretical formulations of shell model accounting for the trough-the-thickness stretching, which allows for large deformations and direct use of 3d constitutive equations. Three different possibilities for implementing this model within the framework of the finite element method are examined, one leading to 7 nodal parameters and the remaining two to 6 nodal parameters. Comparisons are performed of the 7- parameter shell model with no simplification of kinematic terms and 7-parameter shell model which exploits usual simplifications of Green-Lagrange strain measures. Comparisons are also presented of two different ways of implementing the incompatible mode method for reducing the number of shell model parameters to 6. One implementation uses additive decomposition of the strains and the other additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein.

On enhanced strain methods for small and finite deformations of solids

Numerical simulations of engineering problems require robust elements. For a broad range of applications these elements should perform well in bending dominated situations and also in cases of incompressibility. The element should be insensitive against mesh distortions which frequently occur due to modern mesh generation tools or during finite deformations. Possibly the elements should not lock in the thin limits and thus be applicable to shell problems. Furthermore due to efficiency reasons a good coarse mesh accuracy is required in nonlinear analysis. In this paper we discuss the family of enhanced strain elements in order to depict the positive and negative aspects related to these elements. Throughout this discussion we use numerical examples to underline the theoretical results.

IMPROVED ENHANCED STRAIN FOUR‐NODE ELEMENT WITH TAYLOR EXPANSION OF THE SHAPE FUNCTIONS

A class of enhanced strain four-node elements with Taylor expansion of the shape function derivatives is presented. A new concept of enhancement using besides the ‘standard’ enhanced strain fields also two other enhanced fields is developed on the basis of the Hu–Washizu principle. For first-order Taylor expansion enhanced modes become uncoupled, thus only a negligible amount of computing effort for the static condensation of enhanced modes is needed. Furthermore, the formulation permits a symbolic integration, which leads to a closed-form solution for the element tangent matrix. Several numerical examples show that the element is stable, invariant, passes the patch test and yields good results especially in the highly distorted regime.

Shape sensitivity analysis of large deformation frictional contact problems

Sensitivity analysis of large displacement multi-body two-dimensional contact problems with friction is developed in the paper. The incremental (path-dependent) sensitivity problem is derived by direct differentiation of the discretized equations governing the direct problem. In view of finite deformations, due attention is paid to spatial and nominal contact tractions and to proper formulation of the contact laws within the penalty approach. For these reasons an extended node-to-segment contact element is used to model the frictional contact interactions. As the finite elasto-plastic deformations of the contacting bodies are considered, the numerical procedures for computation of all the necessary characteristic formulae of the solid elements (for both the direct and the sensitivity problem) are automatically derived and generated using the symbolic algebra package AceGen. Numerical examples of shape and parameter sensitivity analysis illustrate the approach.

Automation of Finite Element Methods

New finite elements are needed as well in research as in industry environments for the development of virtual prediction techniques. The design and implementation of novel finite elements for specific purposes is a tedious and time consuming task, especially for nonlinear formulations. The automation of this process can help to speed up this process considerably since the generation of the final computer code can be accelerated by order of several magnitudes. This book provides the reader with the required knowledge needed to employ modern automatic tools like AceGen within solid mechanics in a successful way. It covers the range from the theoretical background, algorithmic treatments to many different applications. The book is written for advanced students in the engineering field and for researchers in educational and industrial environments

Dynamics and time-stepping schemes for elastic shells undergoing finite rotations

In this work we study the time-stepping schemes for shell models, which describe the shell-director vector motion by the finite rotations. Different possibilities for choosing director rotations are examined and their relationships are cast in terms of the commutative diagram. The Newmark time-stepping schemes, making use of different rotation parameters, are then developed. The mid-point scheme modified to either conserve or dissipate the total energy is further examined. Several numerical simulations are presented to illustrate the performance of each developed scheme.

Efficient Low Order Virtual Elements for Anisotropic Materials at Finite Strains

Virtual elements were introduced in the last decade and applied to problems in solid mechanics. The success of this methodology when applied to linear problems asks for an extension to the nonlinear regime. This work is concerned with the numerical simulation of structures made of anisotropic material undergoing large deformations. Especially problems with hyperelastic matrix materials and transversly isotropic behaviour will be investigated. The virtual element formulation is based on a low-order formulations for problems in two dimensions. The elements can be arbitrary polygons. The formulation considered relies on minimization of energy, with a novel construction of the stabilization energy and a mixed approximation for the fibers describing the anisotropic behaviour. The formulation is investigated through a several numerical examples, which demonstrate their efficiency, robustness, convergence properties, and locking-free behaviour.

Basic Equations of Continuum Mechanics

This chapter contains a summary of the continuum mechanics background that is needed for the finite element formulation of solid mechanics and structural problems. In detail the kinematical relations and the balance laws with their weak forms are described in this chapter.

Automation of Sensitivity Analysis

Optimization of solid structures lead to better engineering designs and thus to a longer life time of e.g. technical components, engines and buildings. In an optimization process as well material parameters as shapes can be improved. However the computations needed to find optimal solutions are quite complex. Here a proper sensitivity analysis provides an efficient tool in order to compute complex derivations that are needed in the associated algorithms which finally leads to faster convergence of the underlying iterative procedures. Another applications where sensitivity analysis is needed is related to multi-scale analysis of coplex structures. Especially the so called FE\({}^2\)-schemes need an accurate computation of sensitivities. Here micro and macro levels are combined in the computations to account for complex material behaviour at micro level.