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Wikipedia / Interval finite element

In numerical analysis, the interval finite element method (interval FEM) is a finite element method that uses interval parameters. Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of the structure. This is important in concrete structures, wood structures, geomechanics, composite structures, biomechanics and in many other areas. The goal of the Interval Finite Element is to find upper and lower bounds of different characteristics of the model (e.g. stress, displacements, yield surface etc.) and use these results in the design process. This is so called worst case design, which is closely related to the limit state design.

Worst case design requires less information than probabilistic design however the results are more conservative [Köylüoglu and Elishakoff 1998].

Applications of the interval parameters to the modeling of uncertainty

where a and b are real numbers.

Very often, the exact values of the parameters a and b are unknown.

There are several definitions of the solution set of this equation with interval parameters.

United solution set

This is the most popular solution set of the interval equation and this solution set will be applied in this article.

In the multidimensional case the united solutions set is much more complicated. The solution set of the following system of linear interval equations is shown on the following picture

The exact solution set is very complicated, thus it is necessary to find the smallest interval which contains the exact solution set See also http://www.ippt.gov.pl/~kros/pccmm99/01Title.html

Parametric solution set of interval linear system

The Interval Finite Element Method requires the solution of a parameter-dependent system of equations (usually with a symmetric positive definite matrix.) An example of the solution set of general parameter dependent system of equations

is shown on the picture below.

Algebraic solution

In this approach x is an interval number

, then the solution doesn't exist. It is very complex to find a physical interpretation of the algebraic interval solution set. Thus, in applications, the united solution set is usually applied.

The method

Consider the PDE with the interval parameters

Solution of the equation (

where is a stiffness matrix and is a right hand side.

In the simplest case above system can be treat as a system of linear interval equations.

Interval solution versus probabilistic solution

It is important to know that the interval parameters generate different results than uniformly distributed random variables.

Calculations of probabilistic characteristics require the knowledge of a lot of experimental results.

times wider than the sum of appropriate normally distributed random variables.

Sum of n

Consider normally distributed random variable X

Sum of n normally distributed random variable is a normally distributed random variable with the following characteristics (see Six Sigma

We can assume that the width of the probabilistic result is equal to 6 sigma (compare Six Sigma

Because of that the results of the interval finite element (or in general worst-case analysis) may be overestimated in comparison to the stochastic fem analysis (see also propagation of uncertainty). However, in the case of nonprobabilistic uncertainty it is not possible to apply pure probabilistic methods. Because probabilistic characteristic in that case are not known exactly (Elishakoff 2000).

It is possible to consider random (and fuzzy random variables) with the interval parameters (e.g. with the interval mean, variance etc.). Some researchers use interval (fuzzy) measurements in statistical calculations (e.g. http://www.cs.utep.edu/interval-comp/interval.02/fers.pdf ). As a results of such calculations we will get so called imprecise probability.

Imprecise probability is understood in a very wide sense. It is used as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities. It includes both qualitative (comparative probability, partial preference orderings, ...) and quantitative modes (interval probabilities, belief functions, upper and lower previsions, ...). Imprecise probability models are needed in inference problems where the relevant information is scarce, vague or conflicting, and in decision problems where preferences may also be incomplete https://www.sipta.org/.

Simple example: modeling tension, compression, strain, and stress)

1-dimension example

In the tension-compression problem, the following equation shows the relationship between displacement and force where is length, is the area of a cross-section, and is Young's modulus.

To find upper and lower bounds of the displacement , calculate the following partial derivatives

Calculate strain

The same methodology can be applied to the stress

The example is very simple but it shows the applications of the interval parameters in mechanics. Interval FEM use very similar methodology in multidimensional cases [Pownuk 2004].

However, in the multidimensional cases relation between the uncertain parameters and the solution is not always monotone. In those cases, more complicated optimization methods have to be applied.

Multidimensional example

In the case of tension-compression where is displacement, is Young's modulus, is an area of cross-section, and

If Young's modulus and are uncertain then the interval solution can be defined in the following way

For each FEM element it is possible to multiply the equation by the test function

After integration by parts

left endpoint of the element number "e". Approximate solution in the "e"-th element is a linear combination of the shape functions

After substitution to the weak form of the equation we will get the following system of equations

is the right hand side.

In the case of tension-compression problem

If we neglect the distributed load

After taking into account the boundary conditions the stiffness matrix has the following form

Right-hand side has the following form

Let's assume that Young's modulus , area of cross-section and the load

The interval solution can be defined calculating the following way

is in general NP-hard, however in specific cases it is possible to calculate the solution which can be used in many engineering applications.

The uncertain system is safe if the interval solution satisfy all safety conditions.

In postprocessing it is possible to calculate the interval stress, the interval strain and the interval limit state functions and use these values in the design process.

The interval finite element method can be applied to the solution of problems in which there is not enough information to create reliable probabilistic characteristic of the structures (Elishakoff 2000). Interval finite element method can be also applied in the theory of imprecise probability.

Endpoints combination method

. . Upper and lower bound of the solution can be calculated in the following way

Endpoints combination method gives solution which is usually exact; unfortunately the method has exponential computational complexity and cannot be applied to the problems with many interval parameters.

Taylor expansion method

can be expanded by using Taylor series. In the simplest case the Taylor series use only linear approximation

Upper and lower bound of the solution can be calculated by using the following formula

The method is very efficient however it is not very accurate. In order to improve accuracy it is possible to apply higher order Taylor expansion [Pownuk 2004]. This approach can be also applied in the interval finite difference method and the interval boundary element method.

Gradient method

is monotone and the exact solution can be calculated very fast.

Extreme values of the solution can be calculated in the following way

In many structural engineering applications the method gives exact solution. If the solution is not monotone the solution is usually reasonable. In order to improve accuracy of the method it is possible to apply monotonicity tests and higher order sensitivity analysis. The method can be applied to the solution of linear and nonlinear problems of computational mechanics [Pownuk 2004]. Applications of sensitivity analysis method to the solution of civil engineering problems can be found in the following paper [M.V. Rama Rao, A. Pownuk and I. Skalna 2008]. This approach can be also applied in the interval finite difference method and the interval boundary element method.

Element by element method

Muhanna and Mullen applied element by element formulation to the solution of finite element equation with the interval parameters. Using that method it is possible to get the solution with guaranteed accuracy in the case of truss and frame structures.

Perturbation methods

stiffness matrix can be expanded by using perturbation theory. Perturbation theory lead to the approximate value of the interval solution. The method is very efficient and can be applied to large problems of computational mechanics.

Response surface method

by using response surface. Then it is possible to use the response surface to the get the interval solution. Using response surface method it is possible to solve very complex problem of computational mechanics.

Pure interval methods

Several authors tried to apply pure interval methods to the solution of finite element problems with the interval parameters. In some cases it is possible to get very interesting results e.g. [Popova, Iankov, Bonev 2008]. However, in general the method generates very overestimated results.

Parametric interval systems

Popova and Skalna introduced the methods for the solution of the system of linear equations in which the coefficients are linear combinations of interval parameters. In this case it is possible to get very accurate solution of the interval equations with guaranteed accuracy.

References

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