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Pragmatism and the Finite Element Me...

Shakeri, Rezgar,

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Pragmatism and the Finite Element Method in the Modern Era /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Pragmatism and the Finite Element Method in the Modern Era // Rezgar Shakeri.
Author:	Shakeri, Rezgar,
Description:	1 electronic resource (144 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-11, Section: B.
Contained By:	Dissertations Abstracts International85-11B.
Subject:	Applied mathematics. -
Online resource:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31235681
ISBN:	9798382719979

Pragmatism and the Finite Element Method in the Modern Era /
Shakeri, Rezgar,

Pragmatism and the Finite Element Method in the Modern Era /Rezgar Shakeri. - 1 electronic resource (144 pages)

Source: Dissertations Abstracts International, Volume: 85-11, Section: B.

Since numerical modeling of hyperelastic materials is ubiquitous in many engineering fields, a main effort in computational mechanics is to develop stable, accurate and efficient tools for their analysis. A stable numerical calculation of a well-conditioned function will have a relative accuracy of εmachine. Many open source finite element analysis packages contain numerically unstable formulations at some point, even for simple material models. We trace the source of this instability and show how to formulate various compressible/incompressible hyperelastic constitutive models in a stable way.For an incompressible material, the hydrostatic stress becomes an independent variable. Thus, it can be discretized independently of the displacement field. One of the common approaches of simulating the incompressible model is mixed finite element discretization. Mixed finite element method is an efficient approach to overcome locking that is observed in the numerical treatment of nearly incompressible materials when a pure displacement method with linear element is used. In order to guarantee the stability and optimal convergence of a mixed formulation, the inf-sup conditions must be satisfied. The inf-sup constants, which are computable through a set of eigenvalue problems for a given mixed discretization, determines the stability of the chosen FE space for displacement and pressure fields.Despite the accuracy gained by high-order mixed finite element method, most of the industrial finite element problems are constructed within the framework of matrix-based approach by using at most quadratic/linear order of interpolation for displacement/pressure fields. A matrix-free approach yields better performance, both in terms of storage and solve time over the assembled sparse matrices method. We introduce a matrix-free p-multigrid mixed finite element discretization with Newton-Krylov iterative solvers for the hyperelasticity problem. We implemented our problem in Ratel which is a performance portable solid mechanics library that uses matrix-free operators from libCEED and solvers from PETSc to provide fast, efficient, and accurate simulations on next-generation architectures. To demonstrate the reliability and efficiency of the mixed high-order finite element implementation, a series of numerical examples are investigated with p-multigrid preconditioning and different interpolation orders.

English

ISBN: 9798382719979Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

High-order discretization

Pragmatism and the Finite Element Method in the Modern Era /
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500 $a Advisors: Regueiro, Richard A. Committee members: Brown, Jed; Song, Jeong-Hoon; Zhang, Yida; MacCurdy, Robert.
502 $b Ph.D. $c University of Colorado at Boulder $d 2024.
520 $a Since numerical modeling of hyperelastic materials is ubiquitous in many engineering fields, a main effort in computational mechanics is to develop stable, accurate and efficient tools for their analysis. A stable numerical calculation of a well-conditioned function will have a relative accuracy of εmachine. Many open source finite element analysis packages contain numerically unstable formulations at some point, even for simple material models. We trace the source of this instability and show how to formulate various compressible/incompressible hyperelastic constitutive models in a stable way.For an incompressible material, the hydrostatic stress becomes an independent variable. Thus, it can be discretized independently of the displacement field. One of the common approaches of simulating the incompressible model is mixed finite element discretization. Mixed finite element method is an efficient approach to overcome locking that is observed in the numerical treatment of nearly incompressible materials when a pure displacement method with linear element is used. In order to guarantee the stability and optimal convergence of a mixed formulation, the inf-sup conditions must be satisfied. The inf-sup constants, which are computable through a set of eigenvalue problems for a given mixed discretization, determines the stability of the chosen FE space for displacement and pressure fields.Despite the accuracy gained by high-order mixed finite element method, most of the industrial finite element problems are constructed within the framework of matrix-based approach by using at most quadratic/linear order of interpolation for displacement/pressure fields. A matrix-free approach yields better performance, both in terms of storage and solve time over the assembled sparse matrices method. We introduce a matrix-free p-multigrid mixed finite element discretization with Newton-Krylov iterative solvers for the hyperelasticity problem. We implemented our problem in Ratel which is a performance portable solid mechanics library that uses matrix-free operators from libCEED and solvers from PETSc to provide fast, efficient, and accurate simulations on next-generation architectures. To demonstrate the reliability and efficiency of the mixed high-order finite element implementation, a series of numerical examples are investigated with p-multigrid preconditioning and different interpolation orders.
546 $a English
590 $a School code: 0051
650 4 $a Applied mathematics. $3 2122814
650 4 $a Computer science. $3 523869
650 4 $a Architectural engineering. $3 3174102
650 4 $a Environmental engineering. $3 548583
653 $a High-order discretization
653 $a Incompressible materials
653 $a Matrix-free operators
653 $a Numerical stability
653 $a Hyperelastic materials
690 $a 0543
690 $a 0364
690 $a 0984
690 $a 0775
690 $a 0462
710 2 $a University of Colorado at Boulder. $b Civil, Environmental, and Architectural Engineering. $e degree granting institution. $3 3765764
720 1 $a Regueiro, Richard A. $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 85-11B.
790 $a 0051
791 $a Ph.D.
792 $a 2024
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31235681