A Verified Monotonicity‐Based Solution of a Simple Finite Element Model with Uncertain Node Locations

Smith, Andrew P.; Garloff, Jürgen; Werkle, Horst

doi:10.1002/pamm.201010071

Cited by 3 publications

(7 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…xà a x , meaning that interval arithmetic assumes that uncertain variables are treated as unknown, but bounded [57]. A description and the algorithms of the basic operations (addition, subtraction, multiplication, and division) -which naturally differ from operations defined for real numbers -can be found in [24].…”

Section: Interval Methodsmentioning

confidence: 99%

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

Pełczyński

2018

Archives of Civil Engineering

View full text Add to dashboard Cite

Civil engineering is one of the many fields of occurrences of uncertain parameters. The present paper in an attempt to present and describe the most common methods used for inclusions of uncertain parameters. These methods can be applied in the area of civil engineering as well as for a larger domain. Definitions and short explanations of methods based on probability, interval analysis, fuzzy sets, and convex sets are presented. Selected advantages, disadvantages, and the most common fields of implementation are indicated.An example of a cantilever beam presented in this paper shows the main differences between the methods. Results of the performed analysis indicate that the use of convex sets allows us to obtain an accuracy of results similar to stochastic models. At the same time, the computational speed characteristic for interval methods is maintained.

show abstract

Section: Interval Methodsmentioning

confidence: 99%

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

Pełczyński

2018

Archives of Civil Engineering

View full text Add to dashboard Cite

show abstract

“…The parametric solution set, if computed without overestimation (which is typically not the case), is the "true" solution set to the problem. An iterative method of solution is detailed in [12] and is employed in [5,16]. As an alternative, global optimisation may be performed, aimed at the minimisation and maximisation of the components of u [6].…”

Section: Solution Of Interval Systems Of Equationsmentioning

confidence: 99%

“…A downward loading force of 50kN is separately applied to both nodes 4 and 5. This is adapted from the model in [16] by the addition of an extra element, making it no longer statically-determinate. Upon loading, we wish to compute the displacements of nodes 3-5, viz.…”

Section: A Simple Examplementioning

confidence: 99%

“…Typically, more advanced models involve polynomial or rational parameter dependencies, in which case the coefficients of the systems of linear equations to be solved are polynomial or rational functions of the parameters. In [5,16] we present approaches to solve such systems, employing a generalpurpose fixed-point iteration using interval arithmetic [12], an efficient method for bounding the range of a multivariate polynomial over a given box based on the expansion of this polynomial into Bernstein polynomials [4,15], and interval tightening methods. Most of the problems treated in the cited works only exhibit uncertainty in either the material values or the loading forces.…”

Section: Introductionmentioning

confidence: 99%

“…However, in real-life problems, not only the lengths are uncertain but also the positions of the nodes are not exactly known. A statically-determinate problem with uncertain node locations was successfully solved in [16].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

GS_232 — Structural reliability (2)

2011

Applications of Statistics and Probability in Civil Engineering

View full text Add to dashboard Cite

Many problems in structural mechanics are solved using the finite element method (FEM), wherein a model for a mechanical system is constructed by discretising the structure into a finite set of structural elements, connected at nodes, leading to a system of equations to be solved. In the case of linearised geometric displacement equations and linear elastic material behaviour, a system of linear equations is obtained. There may be uncertainty in some or all of the physical model parameters, caused for example by measurement and fabrication imprecision, round-off errors, and various other kinds of inexact knowledge. Intervals can be used to model such parameters when their values are known to lie within certain bounds. In this case, we obtain a system of equations involving interval parameters. However, a naive solution of this system, using interval arithmetic, will typically lead to a solution with result intervals that are hopelessly wide. Previous work has dealt with models where the parameters for uncertain material values (e.g. the Young's modulus and elements' cross-sectional areas) are intervals. However a mechanical frame or truss structure will typically be constructed so that the node positions, before loading, are only known to a tolerance of several millimetres. In this work, we therefore consider not only uncertain material parameters but also uncertain node locations and correspondingly uncertain element lengths, as well as uncertain loading forces. In our approach, firstly guaranteed starting interval enclosures for the node displacements which are relatively wide are computed. These solution intervals are then iteratively tightenend by performing a monotonicity analysis of all the parameters coupled with a solver for interval systems of linear equations. In this way it is possible to provide tight guaranteed enclosures for the node displacements of the model. A simple truss model is presented by way of illustration.

show abstract

Application of the Theory of Convex Sets for Engineering Structures with Uncertain Parameters

Pełczyński

2020

Applied Sciences

View full text Add to dashboard Cite

The present paper discusses an innovative approach providing the solution sets of engineering structures with uncertain parameters. The approach is based on the properties of convex sets and can be applied to structures described by the system of algebraic equations. The present paper focuses on trusses and frames applications, but in general it can be applied to various structures made of thin and thick bars and some plate and shell problems. The uncertain parameters are assumed to be independent. In addition, calculations are valid for any level of uncertainty and the obtained solution sets are exact within the assumed theory and are insensitive for perturbed data. Furthermore, solutions obtained by the present approach can be considered as benchmark solutions and can be used as a reference for other algorithms. The presented formulae allow the analysis of the influence of uncertain parameters on the behaviour of the structure. The presented considerations are illustrated by calculation of two truss examples.

show abstract

A Verified Monotonicity‐Based Solution of a Simple Finite Element Model with Uncertain Node Locations

Abstract: A tight verified solution enclosure is obtained for the node displacements of a simple truss model, whose parameters, including the node locations, are uncertain. The solution is based on a monotonicity analysis of these interval parameters.

Cited by 3 publications

References 3 publications

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

GS_232 — Structural reliability (2)

Application of the Theory of Convex Sets for Engineering Structures with Uncertain Parameters

Contact Info

Product

Resources

About

A Verified Monotonicity‐Based Solution of a Simple Finite Element Model with Uncertain Node Locations

Abstract: A tight verified solution enclosure is obtained for the node displacements of a simple truss model, whose parameters, including the node locations, are uncertain. The solution is based on a monotonicity analysis of these interval parameters.

Cited by 3 publications

References 3 publications

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

On Mathematical Descriptions of Uncertain Parameters in Engineering Structures

GS_232 &#8212; Structural reliability (2)

Application of the Theory of Convex Sets for Engineering Structures with Uncertain Parameters

Contact Info

Product

Resources

About

GS_232 — Structural reliability (2)