In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
, the interval finite element method (interval FEM) is a
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
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
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...
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
Probabilistic design is a discipline within engineering design. It deals primarily with the consideration of the effects of random variability upon the performance of an engineering system during the design phase. Typically, these effects are rel ...
however the results are more conservative
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Applications of the interval parameters to the modeling of uncertainty
Consider the following equation:
where ''a'' and ''b'' are real numbers, and
.
Very often, the exact values of the parameters ''a'' and ''b'' are unknown.
Let's assume that
and
. In this case, it is necessary to solve the following equation
There are several definitions of the solution set of this equation with interval parameters.
United solution set
In this approach the solution is the following 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
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The exact solution set is very complicated, thus it is necessary to find the smallest interval which contains the exact solution set
600px
or simply
where
See als
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.
E. Popova, Parametric Solution Set of Interval Linear System
Algebraic solution
In this approach x is an interval number
In music theory, an interval is a difference in pitch between two sounds.
An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or ha ...
for which the equation
is satisfied. In other words, the left side of the equation is equal to the right side of the equation.
In this particular case the solution is because
If the uncertainty is larger, i.e. , then