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The finite element method (FEM) is the most popular simulation method to predict the physical behaviour of systems and structures. Since analytical solutions are in general not available for most daily problems in engineering sciences numerical methods have been evolved to find a solution for the governing equations of the individual problem. Although the finite element method was originally developed to find a solution for problems of structural mechanics it can nowadays be applied to a large number of engineering disciplines in which the physical description results in a mathematical formulation with some typical differential equations which can be solved numerically.

Much research work has been done in the field of numerical modelling during the last thirty years which enables engineers today to perform simulations close to reality. Nonlinear phenomena in structural mechanics such as nonlinear material behaviour, large deformations or contact problems have become standard modelling tasks. Because of a rapid development in the hardware sector resulting in more and more powerful processors together with decreasing costs of memory it is nowadays possible to perform simulations even for models with millions of degrees of freedom.

In a mathematical sense the finite element solution always just gives one an approximate numerical solution of the considered problem. Sometimes it is not always an easy task for an engineer to decide whether the obtained solution is a good or a bad one. If experimental or analytical results are available it is easily possible to verify any finite element result. However, to predict any structural behaviour in a reliable way without experiments every user of a finite element package should have a certain background about the finite element method in general. In addition, he should have fundamental knowledge about the applied software to be able to judge the appropriateness of the chosen elements and algorithms.

This paper is intended to show a summary of ANSYS capabilities to obtain results of finite element analyses as accurate as possible. Many features of ANSYS are shown and where it is possible we show what is already implemented in ANSYS Workbench.

We will distinguish two different sources of errors within a finite element analysis. On the one hand some mistakes might be introduced in the analysis because the user himself does not know enough about finite elements in general. To minimize these errors we summarize important features of certain element types and element formulations of ANSYS. We also discuss the quality of different element shapes with respect to accuracy and finally provide some information of a correct coupling of different element types. On the other hand errors might occur due to a poor quality of the used finite element program itself. A high-quality program assists the user in reporting reasonable warnings and errors. We will discuss some typical error messages from ANSYS, which allow the user to correct a finite element model immediately. Some reasons possibly leading to poor finite element results are summarized in Tab.1 below to give a coarse overview:

Table 1: Possible sources of errors in a finite element analysis

We should not forget to point out, that ANSYS is a general purpose program, where many numerical modelling techniques are implemented. However, it is sometimes not easy to learn especially for beginners or even for designers, who are usually not finite element experts. With ANSYS Workbench some effort has been done to offer a product, which has implemented by default the best algorithms of ANSYS and is furthermore very easy to use. Hence we will always discuss in this paper as a first step some modelling features of ANSYS and finally point out which of them are already available in ANSYS Workbench.
Preprocessing

First of all, we will summarize some important aspects every user should be familiar with when doing the preprocessing of a finite element analysis with ANSYS. The following topics will be discussed very briefly: consequences of different element shape functions, important features of different beam and shell elements, results of different element shapes and element formulations and finally correct coupling of different element types. In addition to that, we show what kind of help is available from ANSYS in terms of warnings and errors during the preprocessing to minimize modelling mistakes.
User knowledge during preprocessing

Choosing an element with linear or a quadratic shape functions

To make it as easy as possible we will just look at elements with displacement degrees of freedom – no rotational degrees of freedom are present. In ANSYS such elements are called SOLID… (acting in 3D), PLANE…(acting in 2D) or LINK… (acting in 1D), respectively. Let us consider the one-dimensional case. Within one element the displacements are supposed to vary in a linear or quadratic manner:

u(x) = a0 + a1 ⋅ x (1)

u(x) = a0 + a1 ⋅ x + a2 ⋅ x2 (2)

Hence, we talk about linear or quadratic elements. As a consequence of that assumption the strain and also the stress distribution is either constant or linear within each element due to:

ε (x) = du/dx and σ (x) = E ⋅ε(x) (3)

This may be easily illustrated in the following one-dimensional truss example. The resulting displacement and stress distribution is shown for a two element discretization with linear and quadratic elements:

With E as Young’s Modulus, F as applied force, and A(x) as the cross sectional area the analytic solution is

coming from a direct integration of the governing differential equation of the above considered problem. It is generally known that better results can be obtained using the same discretization with quadratic or higher order elements compared with the results of linear or lower order elements. You should also keep in mind that the degree of freedom solution always shows a smooth distribution from element to element whereas the distribution of derived quantities (such as strains and stresses) is no longer smooth at the element boundaries. This is a correct result in terms of the finite element method. Just the so-called “weak formulation” of the concerning differential equation is solved by finite elements and the continuity requirements for the governing variables of the problem are relaxed.

Recommendation:

ANSYS: In general, the user should prefer to take a quadratic element if possible. ANSYS Workbench: By default, SOLID186/SOLID187 is used as a quadratic 3D Solid-Element.

Author: Thomas Nelson, Erke Wang, CADFEM GmbH, Munich, Germany