Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods

G. Rozvany, Tomasz Sokół

Research output: Chapter in Book/Report/Conference proceedingChapter

3 Citations (Scopus)

Abstract

In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed. In the second part of the lecture new and highly efficient numerical methods are discussed, which are capable of optimizing truss topologies with up to over one billion potential truss elements, and thereby providing the verification of optimal volumes with an accuracy of up to four significant digits. The first author will present the first part of this lecture, and the second author the second part.

Original languageEnglish
Title of host publicationCISM International Centre for Mechanical Sciences, Courses and Lectures
PublisherSpringer International Publishing
Pages53-69
Number of pages17
DOIs
Publication statusPublished - Jan 1 2014

Publication series

NameCISM International Centre for Mechanical Sciences, Courses and Lectures
Volume549
ISSN (Print)0254-1971
ISSN (Electronic)2309-3706

Fingerprint

Numerical methods
Exact Solution
Numerical Methods
Topology
Benchmark
Volume Fraction
Volume fraction
Perforated plates
Plane Stress
Structural optimization
Structural Optimization
Topology Optimization
Shape optimization
Digit
Simplification
Analytical Solution
Computing
Zero

Keywords

  • Optimal topology
  • Perforated plate
  • Structural topology optimization
  • Topology simplification
  • Truss element

ASJC Scopus subject areas

  • Mechanical Engineering
  • Mechanics of Materials
  • Computer Science Applications
  • Modelling and Simulation

Cite this

Rozvany, G., & Sokół, T. (2014). Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. In CISM International Centre for Mechanical Sciences, Courses and Lectures (pp. 53-69). (CISM International Centre for Mechanical Sciences, Courses and Lectures; Vol. 549). Springer International Publishing. https://doi.org/10.1007/978-3-7091-1643-2_4

Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. / Rozvany, G.; Sokół, Tomasz.

CISM International Centre for Mechanical Sciences, Courses and Lectures. Springer International Publishing, 2014. p. 53-69 (CISM International Centre for Mechanical Sciences, Courses and Lectures; Vol. 549).

Research output: Chapter in Book/Report/Conference proceedingChapter

Rozvany, G & Sokół, T 2014, Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. in CISM International Centre for Mechanical Sciences, Courses and Lectures. CISM International Centre for Mechanical Sciences, Courses and Lectures, vol. 549, Springer International Publishing, pp. 53-69. https://doi.org/10.1007/978-3-7091-1643-2_4
Rozvany G, Sokół T. Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. In CISM International Centre for Mechanical Sciences, Courses and Lectures. Springer International Publishing. 2014. p. 53-69. (CISM International Centre for Mechanical Sciences, Courses and Lectures). https://doi.org/10.1007/978-3-7091-1643-2_4
Rozvany, G. ; Sokół, Tomasz. / Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. CISM International Centre for Mechanical Sciences, Courses and Lectures. Springer International Publishing, 2014. pp. 53-69 (CISM International Centre for Mechanical Sciences, Courses and Lectures).
@inbook{30494fb4a7bc47e58a08a583704bc328,
title = "Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods",
abstract = "In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed. In the second part of the lecture new and highly efficient numerical methods are discussed, which are capable of optimizing truss topologies with up to over one billion potential truss elements, and thereby providing the verification of optimal volumes with an accuracy of up to four significant digits. The first author will present the first part of this lecture, and the second author the second part.",
keywords = "Optimal topology, Perforated plate, Structural topology optimization, Topology simplification, Truss element",
author = "G. Rozvany and Tomasz Sok{\'o}ł",
year = "2014",
month = "1",
day = "1",
doi = "10.1007/978-3-7091-1643-2_4",
language = "English",
series = "CISM International Centre for Mechanical Sciences, Courses and Lectures",
publisher = "Springer International Publishing",
pages = "53--69",
booktitle = "CISM International Centre for Mechanical Sciences, Courses and Lectures",

}

TY - CHAP

T1 - Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods

AU - Rozvany, G.

AU - Sokół, Tomasz

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed. In the second part of the lecture new and highly efficient numerical methods are discussed, which are capable of optimizing truss topologies with up to over one billion potential truss elements, and thereby providing the verification of optimal volumes with an accuracy of up to four significant digits. The first author will present the first part of this lecture, and the second author the second part.

AB - In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed. In the second part of the lecture new and highly efficient numerical methods are discussed, which are capable of optimizing truss topologies with up to over one billion potential truss elements, and thereby providing the verification of optimal volumes with an accuracy of up to four significant digits. The first author will present the first part of this lecture, and the second author the second part.

KW - Optimal topology

KW - Perforated plate

KW - Structural topology optimization

KW - Topology simplification

KW - Truss element

UR - http://www.scopus.com/inward/record.url?scp=84922836069&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922836069&partnerID=8YFLogxK

U2 - 10.1007/978-3-7091-1643-2_4

DO - 10.1007/978-3-7091-1643-2_4

M3 - Chapter

AN - SCOPUS:84922836069

T3 - CISM International Centre for Mechanical Sciences, Courses and Lectures

SP - 53

EP - 69

BT - CISM International Centre for Mechanical Sciences, Courses and Lectures

PB - Springer International Publishing

ER -