Interval finite element
Interval FEM can be applied in situations where it is not possible to get reliable probabilistic characteristics of the structure.
[1] 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.)
See also [1] 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
It is very complex to find a physical interpretation of the algebraic interval solution set.
is very complicated because of that in practice it is more interesting to find the smallest possible interval which contain the exact solution set
Finite element method lead to the following parameter dependent system of algebraic equations
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.
However, in the case of nonprobabilistic uncertainty it is not possible to apply pure probabilistic methods.
Some researchers use interval (fuzzy) measurements in statistical calculations (e.g. [2] Archived 2010-06-16 at the Wayback Machine).
It is used as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities.
In the tension-compression problem, the following equation shows the relationship between displacement u and force P:
To find upper and lower bounds of the displacement u, calculate the following partial derivatives:
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.
where u is displacement, E is Young's modulus, A is an area of cross-section, and n is a distributed load.
In order to assemble the global stiffness matrix it is necessary to consider an equilibrium equations in each node.
Let's assume that Young's modulus E, area of cross-section A and the load P are uncertain and belong to some intervals
is in general NP-hard, however in specific cases it is possible to calculate the solution which can be used in many engineering applications.
Let's assume that the displacements in the column have to be smaller than some given value (due to safety).
The uncertain system is safe if the interval solution satisfy all safety conditions.
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.
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.
[4] Using that method it is possible to get the solution with guaranteed accuracy in the case of truss and frame structures.
Perturbation theory lead to the approximate value of the interval solution.
[6] Using response surface method it is possible to solve very complex problem of computational mechanics.
In this case it is possible to get very accurate solution of the interval equations with guaranteed accuracy.
