Simulation Technology FAQ

The finite element method (FEM) provides an approximate solution. In engineering practice it is important to know not only the information one wishes to compute but also to have an indication about the size of error of FEM approximation. The p-version FEM makes it convenient and efficient to obtain error estimates in terms of the data of interest.

Research on the p-version dates back to the late 1960’s. Many important advances occurred in the 1970’s. The theoretical basis of the p-version was established in 1981 and optimal meshing strategies appropriate for the p-version were developed in the period 1984-1985. For details, refer to B. Szabó and I. Babuška. Finite Element Analysis. John Wiley & Sons, Inc. (1991). Beginning in 1985, these developments were made available for use in professional practice. The p-version FEM is a more recent technology than the h-version, the development of which began in the late 1950’s.

Yes. For typical design problems in mechanical, aerospace and structural engineering practice the errors of approximation are reduced at an exponential rate when the number of degrees of freedom are increased, provided that the finite element mesh is properly constructed. The h-version can provide algebraic convergence rates only. This makes error control much more effective in the p-version. Furthermore, a converging sequence of solutions is much more naturally and conveniently obtained with the p-version than with the h-version. This makes it feasible to employ quality control procedures in the setting of practical engineering decision-making processes.

Simulation governance is the exercise of command and control over all aspects of simulation. For details please refer to B. Szabó and R. Actis. Simulation governance: Technical requirements for mechanical design. Computer Methods in Applied Mechanics and Engineering. Vol. 249-252, pp. 158 – 168, 2012.

A mathematical model is a transformation (D, I) → F where D represents the available data D that characterize the physical reality we wish to simulate, I represents our understanding (idea) of how to describe that physical reality by mathematical means, and F represents the data of interest such as stress, flux density, natural frequency, etc. that we need for predicting the response of a physical system to some form of excitation.

For example, an aerospace engineer may be interested in predicting the fatigue life of a critical airframe component, such as a bulkhead. In this case D includes a detailed geometrical description of the bulkhead in the form of a CAD model, the material properties, including statistical data of the relationship between stress and the number of cycles to failure, and the design load spectra. The engineer’s experience and understanding of how to transform these data into the predicted number of cycles to failure is represented by I. The transformation involves solving a problem of elasticity to determine the location and magnitude of the maximum normal stress and one or more models of damage accumulation. The data of interest (F) are the probabilities of failure at various locations within the bulkhead, given, for example, the number of ground-air-ground cycles.

Yes. A mathematical model must be formulated so as to take into account all factors that significantly influence the data of interest F. Furthermore, the transformation (D, I) → F must be valid for all D under consideration and the formulation must be such that small changes in D will cause small changes in F. The precisely stated technical requirement is that the transformation must be continuous with respect to the input data.

Because mathematical models used in engineering practice tend to be very complicated, the best one can do is to approximate the data of interest F by numerical means. The numerical approximation to F is denoted by F_num.

In the example of fatigue life prediction described above, the numerical solution includes stress analysis of the bulkhead by the finite element method or some other numerical method, as well as statistical computations such as application of the Monte Carlo method.

For additional details and references please refer to Selected References.

Although the term “finite element model” is frequently used, this term confuses rather than enlightens. This is because it mixes two fundamentally different entities: The mathematical model and the method (specifically the finite element method) used for approximating the solution of the mathematical model. Please refer to ESRD’s technical brief entitled “ Why Finite Element Modeling is NOT Numerical Simulation?” as well as notes on finite element modeling.

The qualifier “physics-based” is superfluous. Think what an alternative would be. Would it be “metaphysics-based model”? There are various ways of describing physical reality by mathematical means at the macroscopic and microscopic levels. The formulation of a mathematical model depends on our understanding of how to represent physical reality by mathematical means and the intended use of the model.

The following statement by Wolfgang Pauli (Nobel Prize, Physics, 1945) captures the essentials remarkably well: The layman always means, when he says “reality” that he is speaking of something self-evidently known; whereas to me it seems the most important and exceedingly difficult task of our time is to work on the construction of a new idea of reality.

Virtual experimentation is a process by which the sensitivity of the data of interest with respect to assumptions incorporated in a mathematical model is tested.

For example, in linear elasticity a linear relationship is assumed to exist between stress and strain and the equations of equilibrium are written with respect to undeformed configuration, implying the assumption that the difference between the deformed and undeformed configurations is negligible. These assumptions are justifiable only under certain conditions. Analysts need to verify that the solution of a mathematical model is consistent with the underlying assumptions.

For instance, if the mathematical model is based on small-strain and small-deformation theory of elasticity then the deformed configuration plotted on 1:1 scale should be almost indistinguishable from the undeformed configuration. If the difference is substantial then a mathematical model based on large-deformation should be used. Should the equivalent stress (von Mises stress) exceed the yield stress, it will be necessary to solve a nonlinear problem taking plastic deformation into account. This raises additional questions such as which model of plasticity should be used? How sensitive are the data of interest to the choice of plasticity models and the parameters that characterize plastic deformation? These questions can be answered through virtual experimentation.

The hierarchic modeling framework implemented in StressCheck was designed to support virtual experimentation.

Suppose that you are a strength engineer in an aerospace company. Someone suggests that since you do not know the service loads accurately, you should not worry about the accuracy of the numerical solution obtained by the finite element method. How would you respond?

It is true that service loads acting on an airframe are not known accurately. However, as a strength engineer you were given the task to consider very precisely defined static and dynamic loads, prescribed by the certification criteria to determine margins of safety. Therefore you know the loads that the airframe is expected to resist very accurately. Since your assignment is to satisfy criteria for certification, you need not worry about uncertainties in the service loads. However you must provide evidence that the error of approximation is within acceptable tolerances.

No. The equilibrium of nodal forces has nothing to do with the finite element solution. The nodal forces will be always in equilibrium even if the error of approximation is very large. A detailed explanation is available in Szabó B. and Babuška, I. Introduction to Finite Element Analysis: Formulation, Verification and Validation. John Wiley & Sons Ltd. 2011, Section 7.2.

The assumption that the strain is small is incorporated in the model by neglecting the nonlinear terms of the strain tensor. No restriction is imposed on the size of the linear strain terms. Furthermore the stress – strain law is linear, independent of the strain. At re-entrant edges and vertices, abrupt changes in boundary conditions and material properties, the strain can become infinitely large. Of course, this is physically impossible. All mathematical models are based on certain assumptions that analysts must clearly understand and consider in interpreting the data based on models. Mathematical models should never be confused with the physical reality they are supposed to represent.