Interval finite element

Consider the following equation:

$${\displaystyle ax=b}$$

where a and b are real numbers, and ${\displaystyle x={\frac {b}{a}}}$.

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

Let's assume that ${\displaystyle a\in [1,2]=\mathbf {a} }$ and ${\displaystyle b\in [1,4]=\mathbf {b} }$. In this case, it is necessary to solve the following equation

$${\displaystyle [1,2]x=[1,4]}$$

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

$${\displaystyle \mathbf {x} =\left\{x:ax=b,a\in \mathbf {a} ,b\in \mathbf {b} \right\}={\frac {\mathbf {b} }{\mathbf {a} }}={\frac {[1,4]}{[1,2]}}=[0.5,4]}$$

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

$${\displaystyle {\begin{bmatrix}{[-4,-3]}&{[-2,2]}\\{[-2,2]}&{[-4,-3]}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}{[-8,8]}\\{[-8,8]}\end{bmatrix}}}$$

is shown on the following picture

$${\displaystyle \sum {_{\exists \exists }}(\mathbf {A} ,\mathbf {b} )=\{x:Ax=b,A\in \mathbf {A} ,b\in \mathbf {b} \}}$$

The exact solution set is very complicated, thus it is necessary to find the smallest interval which contains the exact solution set

$${\displaystyle \diamondsuit \left(\sum {_{\exists \exists }}(\mathbf {A} ,\mathbf {b} )\right)=\diamondsuit \{x:Ax=b,A\in \mathbf {A} ,b\in \mathbf {b} \}}$$

or simply

$${\displaystyle \diamondsuit \left(\sum {_{\exists \exists }}(\mathbf {A} ,\mathbf {b} )\right)=[{\underline {x}}_{1},{\overline {x}}_{1}]\times [{\underline {x}}_{2},{\overline {x}}_{2}]\times \dots \times [{\underline {x}}_{n},{\overline {x}}_{n}]}$$

where

$${\displaystyle {\underline {x}}_{i}=\min\{x_{i}:Ax=b,A\in \mathbf {A} ,b\in \mathbf {b} \},\ \ {\overline {x}}_{i}=\max\{x_{i}:Ax=b,A\in \mathbf {A} ,b\in \mathbf {b} \}}$$

$${\displaystyle x_{i}\in \{x_{i}:Ax=b,A\in \mathbf {A} ,b\in \mathbf {b} \}=[{\underline {x}}_{i},{\overline {x}}_{i}]}$$

See also

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

$${\displaystyle {\begin{bmatrix}p_{1}&p_{2}\\p_{2}+1&p_{1}\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}={\begin{bmatrix}{\frac {p_{1}+6p_{2}}{5.0}}\\2p_{1}-6\end{bmatrix}},\ \ {\text{for}}\ \ p_{1}\in [2,4],p_{2}\in [-2,1].}$$

is shown on the picture below.[2]

Algebraic solution

In this approach x is an interval number for which the equation

$${\displaystyle [1,2]x=[1,4]}$$

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 ${\displaystyle x=[1,2]}$ because

$${\displaystyle ax=[1,2][1,2]=[1,4]}$$

If the uncertainty is larger, i.e. ${\displaystyle a=[1,4]}$, then ${\displaystyle x=[1,1]}$ because

$${\displaystyle ax=[1,4][1,1]=[1,4]}$$

If the uncertainty is even larger, i.e. ${\displaystyle a=[1,8]}$, 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.

Consider the PDE with the interval parameters

$${\displaystyle G(x,u,p)=0}$$

(1)

where ${\displaystyle p=(p_{1},\dots ,p_{m})\in {\mathbf {p} }}$ is a vector of parameters which belong to given intervals

$${\displaystyle p_{i}\in [{\underline {p}}_{i},{\overline {p}}_{i}]={\mathbf {p} }_{i},}$$

$${\displaystyle {\mathbf {p} }={\mathbf {p} }_{1}\times {\mathbf {p} }_{2}\times \cdots \times {\mathbf {p} }_{m}.}$$

For example, the heat transfer equation

$${\displaystyle k_{x}{\frac {\partial ^{2}u}{\partial x^{2}}}+k_{y}{\frac {\partial ^{2}u}{\partial y^{2}}}+q=0{\text{ for }}x\in \Omega }$$

$${\displaystyle u(x)=u^{*}(x){\text{ for }}x\in \partial \Omega }$$

where ${\displaystyle k_{x},k_{y}}$ are the interval parameters (i.e. ${\displaystyle k_{x}\in {\mathbf {k} }_{x},\ k_{y}\in {\mathbf {k} }_{y}}$).

Solution of the equation (1) can be defined in the following way

$${\displaystyle {\tilde {u}}(x):=\{u(x):G(x,u,p)=0,p\in {\mathbf {p} }\}}$$

For example, in the case of the heat transfer equation

$${\displaystyle {\tilde {u}}(x)=\left\{u(x):k_{x}{\frac {\partial ^{2}u}{\partial x^{2}}}+k_{y}{\frac {\partial ^{2}u}{\partial y^{2}}}+q=0{\text{ for }}x\in \Omega ,u(x)=u^{*}(x){\text{ for }}x\in \partial \Omega ,k_{x}\in {\mathbf {k} }_{x},\ k_{y}\in {\mathbf {k} }_{y}\right\}}$$

Solution ${\displaystyle {\tilde {u}}}$ is very complicated because of that in practice it is more interesting to find the smallest possible interval which contain the exact solution set ${\displaystyle {\tilde {u}}}$.

$${\displaystyle {\mathbf {u} }(x)=\lozenge {\tilde {u}}(x)=\lozenge \{u(x):G(x,u,p)=0,p\in {\mathbf {p} }\}}$$

For example, in the case of the heat transfer equation

$${\displaystyle {\mathbf {u} }(x)=\lozenge \left\{u(x):k_{x}{\frac {\partial ^{2}u}{\partial x^{2}}}+k_{y}{\frac {\partial ^{2}u}{\partial y^{2}}}+q=0{\text{ for }}x\in \Omega ,u(x)=u^{*}(x){\text{ for }}x\in \partial \Omega ,k_{x}\in {\mathbf {k} }_{x},\ k_{y}\in {\mathbf {k} }_{y}\right\}}$$

Finite element method lead to the following parameter dependent system of algebraic equations

$${\displaystyle K(p)u=Q(p),\ \ \ p\in {\mathbf {p} }}$$

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

Interval solution can be defined as a multivalued function

$${\displaystyle {\mathbf {u} }=\lozenge \{u:K(p)u=Q(p),p\in {\mathbf {p} }\}}$$

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

It is also possible to define the interval solution as a solution of the following optimization problem

$${\displaystyle {\underline {u}}_{i}=\min\{u_{i}:K(p)u=Q(p),p\in {\mathbf {p} }\}}$$

$${\displaystyle {\overline {u}}_{i}=\max\{u_{i}:K(p)u=Q(p),p\in {\mathbf {p} }\}}$$

In multidimensional case the interval solution can be written as

$${\displaystyle \mathbf {u} =\mathbf {u} _{1}\times \cdots \times \mathbf {u} _{n}=[{\underline {u}}_{1},{\overline {u}}_{1}]\times \cdots \times [{\underline {u}}_{n},{\overline {u}}_{n}]}$$

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

Interval parameter ${\displaystyle \mathbf {p} =[{\underline {p}},{\overline {p}}]}$ take into account all possible probability distributions (for ${\displaystyle p\in [{\underline {p}},{\overline {p}}]}$).

In order to define the interval parameter it is necessary to know only upper ${\displaystyle {\overline {p}}}$ and lower bound ${\displaystyle {\underline {p}}}$.

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

It is possible to show that the sum of n interval numbers is ${\displaystyle {\sqrt {n}}}$ times wider than the sum of appropriate normally distributed random variables.

Sum of n interval number ${\displaystyle \mathbf {p} =[{\underline {p}},{\overline {p}}]}$ is equal to

$${\displaystyle n\mathbf {p} =[n{\underline {p}},n{\overline {p}}]}$$

Width of that interval is equal to

$${\displaystyle n{\overline {p}}-n{\underline {p}}=n({\overline {p}}-{\underline {p}})=n\Delta p}$$

Consider normally distributed random variable X such that

$${\displaystyle m_{X}=E[X]={\frac {{\overline {p}}+{\underline {p}}}{2}},\sigma _{X}={\sqrt {\operatorname {Var} [X]}}={\frac {\Delta p}{6}}}$$

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

$${\displaystyle E[nX]=n{\frac {{\overline {p}}+{\underline {p}}}{2}},\sigma _{nX}={\sqrt {n\operatorname {Var} [X]}}={\sqrt {n}}\sigma ={\sqrt {n}}{\frac {\Delta p}{6}}}$$

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

$${\displaystyle 6\sigma _{nX}=6{\sqrt {n}}{\frac {\Delta p}{6}}={\sqrt {n}}\Delta p}$$

Now we can compare the width of the interval result and the probabilistic result

$${\displaystyle {\frac {{\text{width of }}n{\text{ intervals}}}{{\text{width of }}n{\text{ random variables}}}}={\frac {n\Delta p}{{\sqrt {n}}\Delta p}}={\sqrt {n}}}$$

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. ). 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 .

It is possible to solve the equation ${\displaystyle K(p)u(p)=Q(p)}$ for all possible combinations of endpoints of the interval ${\displaystyle {\hat {p}}}$.
The list of all vertices of the interval ${\displaystyle {\hat {p}}}$ can be written as ${\displaystyle L=\{p_{1}^{*},...,p_{n}^{*}\}}$.
Upper and lower bound of the solution can be calculated in the following way

$${\displaystyle {\underline {u}}_{i}=\min\{u_{i}(p_{k}^{*}):K(p_{k}^{*})u(p_{k}^{*})=Q(p_{k}^{*}),p_{k}^{*}\in L\}}$$

$${\displaystyle {\overline {u}}_{i}=\max\{u_{i}(p_{k}^{*}):K(p_{k}^{*})u(p_{k}^{*})=Q(p_{k}^{*}),p_{k}^{*}\in L\}}$$

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.[3]

The function ${\displaystyle u=u(p)}$ can be expanded by using Taylor series. In the simplest case the Taylor series use only linear approximation

$${\displaystyle u_{i}(p)\approx u_{i}(p_{0})+\sum _{j}{\frac {\partial u(p_{0})}{\partial p_{j}}}\Delta p_{j}}$$

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

$${\displaystyle {\underline {u}}_{i}\approx u_{i}(p_{0})-\left|\sum _{j}{\frac {\partial u(p_{0})}{\partial p_{j}}}\right|\Delta p_{j}}$$

$${\displaystyle {\overline {u}}_{i}\approx u_{i}(p_{0})+\left|\sum _{j}{\frac {\partial u(p_{0})}{\partial p_{j}}}\right|\Delta p_{j}}$$

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.

If the sign of the derivatives ${\displaystyle {\frac {\partial u_{i}}{\partial p_{j}}}}$ is constant then the functions ${\displaystyle u_{i}=u_{i}(p)}$ is monotone and the exact solution can be calculated very fast.

if ${\displaystyle {\frac {\partial u_{i}}{\partial p_{j}}}\geq 0}$ then ${\displaystyle p_{i}^{\min }={\underline {p}}_{i},\ p_{i}^{\max }={\overline {p}}_{i}}$

if ${\displaystyle {\frac {\partial u_{i}}{\partial p_{j}}}<0}$ then ${\displaystyle p_{i}^{\min }={\overline {p}}_{i},\ p_{i}^{\max }={\underline {p}}_{i}}$

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

$${\displaystyle {\underline {u}}_{i}=u_{i}(p^{\min }),\ {\overline {u}}_{i}=u_{i}(p^{\max })}$$

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.

Muhanna and Mullen applied element by element formulation to the solution of finite element equation with the interval parameters.[4] Using that method it is possible to get the solution with guaranteed accuracy in the case of truss and frame structures.

The solution ${\displaystyle u=u(p)}$ stiffness matrix ${\displaystyle K=K(p)}$ and the load vector ${\displaystyle Q=Q(p)}$ can be expanded by using perturbation theory. Perturbation theory lead to the approximate value of the interval solution.[5] The method is very efficient and can be applied to large problems of computational mechanics.

It is possible to approximate the solution ${\displaystyle u=u(p)}$ by using response surface. Then it is possible to use the response surface to the get the interval solution.[6] Using response surface method it is possible to solve very complex problem of computational mechanics.[7]

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.[8]

Popova[9] and Skalna[10] 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.

• Interval boundary element method
• Interval (mathematics)
• Interval arithmetic
• Imprecise probability
• Multivalued function
• Differential inclusion
• Observational error
• Random compact set
• Reliability (statistics)
• Confidence interval
• Best, worst and average case
• Probabilistic design
• Propagation of uncertainty
• Experimental uncertainty analysis
• Sensitivity analysis
• Perturbation theory
• Continuum mechanics
• Solid mechanics
• Truss
• Space frame
• Linear elasticity
• Strength of materials

• Dempster, A. P. (1967). "Upper and lower probabilities induced by a multivalued mapping". The Annals of Mathematical Statistics 38 (2): 325–339. . Retrieved 2009-09-23
• Analyzing Uncertainty in Civil Engineering, by W. Fellin, H. Lessmann, M. Oberguggenberger, and R. Vieider (eds.), Springer-Verlag, Berlin, 2005
• I. Elishakoff, Possible limitations of probabilistic methods in engineering. Applied Mechanics Reviews, Vol.53, No.2, pp. 19–25, 2000.
• Hlavácek, I., Chleboun, J., Babuška, I.: Uncertain Input Data Problems and the Worst Scenario Method. Elsevier, Amsterdam (2004)
• Köylüoglu, U., Isaac Elishakoff; A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties, Computers & Structures Volume: 67, Issue: 1–3, April 1, 1998, pp. 91–98
• D. Moens and D. Vandepitte, Interval sensitivity theory and its application to frequency response envelope analysis of uncertain structures. Computer Methods in Applied Mechanics and Engineering Vol. 196, No. 21-24,1 April 2007, pp. 2486–2496.
• Möller, B., Beer, M., Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, Berlin, 2004.
• E. Popova, R. Iankov, Z. Bonev: Bounding the Response of Mechanical Structures with Uncertainties in all the Parameters. In R.L.Muhannah, R.L.Mullen (Eds): Proceedings of the NSF Workshop on Reliable Engineering Computing (REC), Svannah, Georgia USA, Feb. 22–24, 2006, 245-265
• A. Pownuk, Numerical solutions of fuzzy partial differential equation and its application in computational mechanics, Fuzzy Partial Differential Equations and Relational Equations: Reservoir Characterization and Modeling (M. Nikravesh, L. Zadeh and V. Korotkikh, eds.), Studies in Fuzziness and Soft Computing, Physica-Verlag, 2004, pp. 308–347
• A. Pownuk, Efficient Method of Solution of Large Scale Engineering Problems with Interval Parameters Based on Sensitivity Analysis, Proceeding of NSF workshop on Reliable Engineering Computing, September 15–17, 2004, Savannah, Georgia, USA, pp. 305–316
• M.V. Rama Rao, A. Pownuk and I. Skalna, Stress Analysis of a Singly Reinforced Concrete Beam with Uncertain Structural Parameters, NSF workshop on Reliable Engineering Computing, February 20–22, 2008, Savannah, Georgia, USA, pp. 459–478
• Bernardini, Alberto, Tonon, Fulvio, Bounding Uncertainty in Civil Engineering, Springer 2010
• Ben-Haim Y., Elishakoff I., 1990, Convex Models of Uncertainty in Applied Mechanics. Elsevier Science Publishers, New York
• Valliappan S., Pham T.D., 1993, Fuzzy Finite Element Analysis of A Foundation on Elastic Soil Medium. International Journal for Numerical and Analytical Methods in Geomechanics, Vol.17, pp. 771–789
• Elishakoff I., Li Y.W., Starnes J.H., 1994, A deterministic method to predict the effect of unknown-but-bounded elastic moduli on the buckling of composite structures. Computer methods in applied mechanics and engineering, Vol.111, pp. 155–167
• Valliappan S. Pham T.D., 1995, Elasto-Plastic Finite Element Analysis with Fuzzy Parameters. International Journal for Numerical Methods in Engineering, 38, pp. 531–548
• Rao S.S., Sawyer J.P., 1995, Fuzzy Finite Element Approach for the Analysis of Imprecisly Defined Systems. AIAA Journal, Vol.33, No.12, pp. 2364–2370
• Köylüoglu H.U., Cakmak A., Nielsen S.R.K., 1995, Interval mapping in structural mechanics. In: Spanos, ed. Computational Stochastic Mechanics. 125–133. Balkema, Rotterdam
• Muhanna, R. L. and R. L. Mullen (1995). "Development of Interval Based Methods for Fuzziness in Continuum Mechanics" in Proceedings of the 3rd International Symposium on Uncertainty Modeling and Analysis and Annual Conference of the North American Fuzzy Information Processing Society (ISUMA–NAFIPS'95),IEEE, 705–710

• Reliable Engineering Computing, Georgia Institute of Technology, Savannah, USA
• Interval Computations
• Reliable Computing (Journal)
• Interval equations (collections of references)
• Interval finite element web applications
• E. Popova, Parametric Solution Set of Interval Linear System
• The Society for Imprecise Probability: Theories and Applications