In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, the interval finite element method (interval FEM) is a
finite element method
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 tran ...
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 Stress (mechanics), stresses. The yield surface is usually convex polytope, convex and the state of stress of ''inside'' the yield surface is elastic. When the stress ...
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
Limit State Design (LSD), also known as Load And Resistance Factor Design (LRFD), refers to a design method used in structural engineering. A limit state is a condition of a structure beyond which it no longer fulfills the relevant design criteri ...
.
Worst case design requires less information than
probabilistic design
Probabilistic design is a discipline within engineering design. It deals primarily with the consideration and minimization of the effects of random variability upon the performance of an engineering system during the design phase. Typically, the ...
however the results are more conservative
öylüoglu and Elishakoff 1998">Elishakoff.html" ;"title="öylüoglu and Elishakoff">öylüoglu and Elishakoff 1998
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
600px
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.
In mathematics
A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon a ...
The exact solution set is very complicated, thus it is necessary to find the smallest interval which contains the exact solution set
600px
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.
In mathematics
A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon a ...
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 har ...
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
The method
Consider the PDE with the interval parameters
where p = (p_1,\dots,p_m) \in is a vector of parameters which belong to given intervals
p_i\in underline p_i,\overline p_i_i,
= _1\times _2 \times \cdots \times _m.
For example, the heat transfer equation
k_x \frac+ k_y\frac +q =0 \text x \in \Omega
u(x)=u^*(x) \text x \in \partial\Omega
where k_x, k_y are the interval parameters (i.e. k_x\in_x, \ k_y\in_y ).
Solution of the equation () can be defined in the following way
\tilde(x):= \
For example, in the case of the heat transfer equation
\tilde(x) = \left\
Solution \tilde is very complicated because of that in practice it is more interesting to find the smallest possible interval which contain the exact solution set \tilde .
(x)=\lozenge \tilde(x) = \lozenge \
For example, in the case of the heat transfer equation
(x) = \lozenge \left\
Finite element method lead to the following parameter dependent system of algebraic equations
K(p) u = Q(p), \ \ \ p \in
where is a stiffness matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
and is a right hand side.
Interval solution can be defined as a multivalued function
= \lozenge \
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
\underline u_i = \min \
\overline u_i = \max \
In multidimensional case the interval solution can be written as
\mathbf = \mathbf_1 \times \cdots \times \mathbf_n = underline u_1,\overline u_1\times \cdots\times underline u_n,\overline u_n
Interval solution versus probabilistic solution
It is important to know that the interval parameters generate different results than uniformly distributed random variables.
Interval parameter \mathbf= underline p,\overline p take into account all possible probability distributions (for p\in underline p,\overline p).
In order to define the interval parameter it is necessary to know only upper \overline p and lower bound \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 \sqrt times wider than the sum of appropriate normally distributed random variables.
Sum of ''n'' interval number \mathbf= underline p,\overline p is equal to
n\mathbf = \underline p,n\overline p
Width of that interval is equal to
n\overline p - n\underline p = n(\overline p - \underline p) = n\Delta p
Consider normally distributed random variable ''X'' such that
m_X=E \frac, \sigma_X=\sqrt=\frac
Sum of ''n'' normally distributed random variable is a normally distributed random variable with the following characteristics (see Six Sigma
Six Sigma (6σ) is a set of techniques and tools for process improvement. It was introduced by American engineer Bill Smith while working at Motorola in 1986.
Six Sigma strategies seek to improve manufacturing quality by identifying and removin ...
)
E Xn\frac, \sigma_=\sqrt=\sqrt\sigma=\sqrt\frac
We can assume that the width of the probabilistic result is equal to 6 sigma (compare Six Sigma
Six Sigma (6σ) is a set of techniques and tools for process improvement. It was introduced by American engineer Bill Smith while working at Motorola in 1986.
Six Sigma strategies seek to improve manufacturing quality by identifying and removin ...
).
6\sigma_=6\sqrt\frac=\sqrt\Delta p
Now we can compare the width of the interval result and the probabilistic result
\frac = \frac = \sqrt
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
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of ex ...
).
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
Isaac Elishakoff is an Israeli-American engineer who is Distinguished Research Professor in the Ocean and Mechanical Engineering Department in the Florida Atlantic University, Boca Raton, Florida. He is an internationally recognized, authoritati ...
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 generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Ther ...
.
Imprecise probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Ther ...
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 incomplet
Simple example: modeling tension, compression, strain, and stress)
image:TensionCompression.JPG, 400px
1-dimension example
In the tension-compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
problem, the following equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
shows the relationship between displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
and force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
:
\fracu = P
where is length, is the area of a cross-section, and is Young's modulus
Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
.
If the Young's modulus and force are uncertain, then
E\in underline E,\overline E P\in underline P,\overline P
To find upper and lower bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of .
Dually, a lower bound or minorant of is defined to be an element of that is less th ...
of the displacement , calculate the following partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s:
\frac = \frac < 0
\frac = \frac > 0
Calculate extreme values of the displacement as follows:
\underline u = u(\overline E,\underline P) = \frac
\overline u = u(\underline E,\overline P) = \frac
Calculate strain using following formula:
\varepsilon = \frac u
Calculate derivative of the strain using derivative from the displacements:
\frac = \frac \frac = \frac < 0
\frac = \frac \frac = \frac > 0
Calculate extreme values of the displacement as follows:
\underline \varepsilon = \varepsilon(\overline E,\underline P) = \frac
\overline \varepsilon = \varepsilon(\underline E,\overline P) = \frac
It is also possible to calculate extreme values of strain using the displacements
\frac = \frac > 0
then
\underline \varepsilon = \varepsilon(\underline u) = \frac
\overline \varepsilon = \varepsilon(\overline u) = \frac
The same methodology can be applied to the stress
\sigma = E \varepsilon
then
\frac = \varepsilon + E\frac =\varepsilon + E\frac \frac = \frac - \frac= 0
\frac = E\frac = E\frac \frac = \frac >0
and
\underline \sigma = \sigma (\underline P) = \frac
\overline \sigma = \sigma (\overline P) = \frac
If we treat stress as a function of strain then
\frac=\frac(E\varepsilon)=E> 0
then
\underline \sigma = \sigma (\underline \varepsilon) =E\underline \varepsilon = \frac
\overline \sigma = \sigma (\overline \varepsilon) = E\overline \varepsilon = \frac
Structure is safe if stress \sigma is smaller than a given value \sigma_0 i.e.,
\sigma < \sigma_0
this condition is true if
\overline \sigma < \sigma_0
After calculation we know that this relation is satisfied if
\frac < \sigma_0
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 ownuk 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
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
problem the equilibrium equation has the following form
\frac\left( EA\frac \right)+n=0
where is displacement, is Young's modulus
Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
, is an area of cross-section, and is a distributed load.
In order to get unique solution it is necessary to add appropriate boundary conditions e.g.
u(0)=0
\left.\frac\_ EA = P
If Young's modulus
Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
and are uncertain then the interval solution can be defined in the following way
(x)=\left\
For each FEM element it is possible to multiply the equation by the test function
\int_^ \left( \frac\left( EA\frac \right)+n \right)v=0
where x \in ,L^
After integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
we will get the equation in the weak form
\int_^ EA\frac \frac dx = \int_^ nv \, dx
where x\in ,L^
Let's introduce a set of grid points x_0, x_1, \dots, x_ , where Ne is a number of elements, and linear shape functions for each FEM element
N_1^(x)=1-\frac, \ \ N_2^(x) = \frac.
where x\in _^, x_^
x_^ left endpoint of the element, x_^ left endpoint of the element number "e".
Approximate solution in the "e"-th element is a linear combination of the shape functions
u^_(x) = u^_1 N_1^(x)+u^_2 N_2^(x), \ \ v^_(x) = u^_1 N_1^(x)+u^_2 N_2^(x)
After substitution to the weak form of the equation we will get the following system of equations
\begin
\frac & -\frac \\
-\frac & \frac \\
\end
\begin u^_1 \\ u^_2 \end
=
\begin
\int_^ n N_1^(x)dx \\
\int_^ n N_2^(x)dx
\end
or in the matrix form
K^ u^ = Q^
In order to assemble the global stiffness matrix it is necessary to consider an equilibrium equations in each node.
After that the equation has the following matrix form
K u = Q
where
K= \begin
K_^ & K_^ & 0 & \cdots & 0 \\
K_^ & K_^+K_^ & K_^ & \cdots & 0 \\
0 & K_^ & K_^+K_^ & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & K_^ + K_^ & K_^ \\
0 & 0 & \cdots & K_^ & K_^
\end
is the global stiffness matrix,
u = \begin
u_0 \\
u_1 \\
\vdots \\
u_ \\
\end
is the solution vector,
Q=\begin
Q_0 \\
Q_1 \\
\vdots \\
Q_ \\
\end
is the right hand side.
In the case of tension-compression problem
K= \begin
\frac & -\frac & 0 & \cdots & 0 \\
-\frac & \frac + \frac & -\frac & \cdots & 0 \\
0 & -\frac & \frac+ \frac & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & \frac + \frac & -\frac \\
0 & 0 & \cdots & -\frac & \frac
\end
If we neglect the distributed load
Q=
\begin
R \\
0 \\
\vdots \\
0 \\
P \\
\end
After taking into account the boundary conditions the stiffness matrix has the following form
K=
\begin
1 & 0 & 0 & \cdots & 0 \\
0 & \frac + \frac & -\frac & \cdots & 0 \\
0 & -\frac & \frac + \frac & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots & \frac + \frac & -\frac \\
0 & 0 & \cdots & -\frac & \frac
\end = K(E,A) = K
Right-hand side has the following form
Q=
\begin
0 \\
0 \\
\vdots \\
0 \\
P \\
\end = Q(P)
Let's assume that Young's modulus , area of cross-section and the load are uncertain and belong to some intervals
E^ \in underline E^,\overline E^
A^ \in underline A^,\overline A^
P \in underline P,\overline P
The interval solution can be defined calculating the following way
\mathbf u = \lozenge \left\
Calculation of the interval vector is in general NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
, however in specific cases it is possible to calculate the solution which can be used in many engineering applications.
The results of the calculations are the interval displacements
u_i \in underline u_i, \overline u_i
Let's assume that the displacements in the column have to be smaller than some given value (due to safety).
u_i< u^_i
The uncertain system is safe if the interval solution satisfy all safety conditions.
In this particular case
u_i< u^_i, \ \ \ u_i\in underline u_i, \overline u_i
or simple
\overline u_i< u^_i
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
Isaac Elishakoff is an Israeli-American engineer who is Distinguished Research Professor in the Ocean and Mechanical Engineering Department in the Florida Atlantic University, Boca Raton, Florida. He is an internationally recognized, authoritati ...
2000). Interval finite element method can be also applied in the theory of imprecise probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Ther ...
.
Endpoints combination method
It is possible to solve the equation K(p)u(p)=Q(p) for all possible combinations of endpoints of the interval \hat p .
The list of all vertices of the interval \hat p can be written as L=\ .
Upper and lower bound of the solution can be calculated in the following way
\underline u_i = \min\
\overline u_i = \max\
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.[A. Neumaier, Interval methods for systems of equations, Cambridge University Press, New York, 1990]
Taylor expansion method
The function u=u(p) can be expanded by using Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
.
In the simplest case the Taylor series use only linear approximation
u_i(p) \approx u_i(p_0)+\sum_j\frac\Delta p_j
Upper and lower bound of the solution can be calculated by using the following formula
\underline u_i \approx u_i(p_0)-\left, \sum_j\frac\\Delta p_j
\overline u_i \approx u_i(p_0)+\left, \sum_j\frac\\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 ownuk 2004
This approach can be also applied in the interval finite difference method and the interval boundary element method.
Gradient method
If the sign of the derivatives \frac is constant then the functions u_i= u_i(p) is monotone and the exact solution can be calculated very fast.
:if \frac \ge 0 then p_i^ = \underline p_i, \ p_i^ = \overline p_i
:if \frac < 0 then p_i^ = \overline p_i, \ p_i^ = \underline p_i
Extreme values of the solution can be calculated in the following way
\underline u_i=u_i(p^), \ \overline u_i=u_i(p^)
In many structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made Structure#Load-bearing, structures. Structural engineers also ...
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
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third wa ...
ownuk 2004 Applications of sensitivity analysis method to the solution of civil engineering problems can be found in the following paper .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.[R.L. Muhanna, R.L. Mullen, Uncertainty in Mechanics Problems - Interval - Based Approach. Journal of Engineering Mechanics, Vol.127, No.6, 2001, 557-556] Using that method it is possible to get the solution with guaranteed accuracy in the case of truss and frame structures.
Perturbation methods
The solution u = u(p) stiffness matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
K = K(p) and the load vector Q = Q(p) can be expanded by using perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. Perturbation theory lead to the approximate value of the interval solution.[Z. Qiu and I. Elishakoff, Antioptimization of structures with large uncertain but non-random parameters via interval analysis Computer Methods in Applied Mechanics and Engineering, Volume 152, Issues 3-4, 24 January 1998, Pages 361-372] The method is very efficient and can be applied to large problems of computational mechanics.
Response surface method
It is possible to approximate the solution u = u(p) by using response surface. Then it is possible to use the response surface to the get the interval solution.[U.O. Akpan, T.S. Koko, I.R. Orisamolu, B.K. Gallant, Practical fuzzy finite element analysis of structures, Finite Elements in Analysis and Design, 38, pp. 93–111, 2000.] Using response surface method it is possible to solve very complex problem of computational mechanics.[M. Beer, Evaluation of Inconsistent Engineering data, The Third workshop on Reliable Engineering Computing (REC08) Georgia Institute of Technology, February 20–22, 2008, Savannah, Georgia, USA.]
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. opova, Iankov, Bonev 2008 However, in general the method generates very overestimated results.[ Kulpa Z., Pownuk A., Skalna I., Analysis of linear mechanical structures with uncertainties by means of interval methods. Computer Assisted Mechanics and Engineering Sciences, vol. 5, 1998, pp. 443–477]
Parametric interval systems
Popova[E. Popova, On the Solution of Parametrised Linear Systems. W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing, Validated Numerics, Interval Methods. Kluwer Acad. Publishers, 2001, pp. 127–138.] and Skalna[I. Skalna, A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters, Reliable Computing, Volume 12, Number 2, April, 2006, pp. 107–120] introduced the methods for the solution of the system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables.
For example,
: \begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of th ...
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.
See also
* Interval boundary element method
* Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real ...
* Interval arithmetic
Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numeri ...
* Imprecise probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Ther ...
* Multivalued function
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional p ...
* Differential inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
:\frac(t)\in F(t,x(t)),
where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...
* Observational error
Observational error (or measurement error) is the difference between a measured value of a quantity and its unknown true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. Such errors are inherent in the measurement ...
* Random compact set In mathematics, a random compact set is essentially a compact set-valued random variable. Random compact sets are useful in the study of attractors for random dynamical systems.
Definition
Let (M, d) be a complete separable metric space. Let \ma ...
* Reliability (statistics)
In statistics and psychometrics, reliability is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions:It is the characteristic of a set of test scores that ...
* Confidence interval
* Best, worst and average case
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, b ...
* Probabilistic design
Probabilistic design is a discipline within engineering design. It deals primarily with the consideration and minimization of the effects of random variability upon the performance of an engineering system during the design phase. Typically, the ...
* Propagation of uncertainty
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of ex ...
* Experimental uncertainty analysis
Experimental uncertainty analysis is a technique that analyses a ''derived'' quantity, based on the uncertainties in the experimentally ''measured'' quantities that are used in some form of mathematical relationship ("model") to calculate that d ...
* Sensitivity analysis
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. This involves estimating sensitivity ...
* Perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
* Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
* Solid mechanics
Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
* Truss
A truss is an assembly of ''members'' such as Beam (structure), beams, connected by ''nodes'', that creates a rigid structure.
In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so ...
* Space frame
In architecture and structural engineering, a space frame or space structure (Three-dimensional space, 3D truss) is a rigid, lightweight, truss-like structure constructed from interlocking struts in a geometry, geometric pattern. Space frames can ...
* Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
* Strength of materials
Strength may refer to:
Personal trait
*Physical strength, as in people or animals
*Character strengths like those listed in the Values in Action Inventory
*The exercise of willpower
Physics
* Mechanical strength, the ability to withstand ...
References
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External links
Reliable Engineering Computing, Georgia Institute of Technology, Savannah, USA
Interval Computations
{{Webarchive, url=https://web.archive.org/web/20080920171250/http://www.cs.utep.edu/interval-comp/ , date=2008-09-20
Reliable Computing (Journal)
E. Popova, Parametric Solution Set of Interval Linear System
The Society for Imprecise Probability: Theories and Applications
Finite element method