Finite Element Solution of Boundary Value Problems

Axelsson, Owe; Barker, V. A.

doi:10.1137/1.9780898719253

Cited by 132 publications

(208 citation statements)

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“…The convergence rate here appears to be O(N −4 ) rather than O(N −2 ): this is not surprising, as it is well known that the finite element method can exhibit nodal superconvergence, when the numerical solution at node points is much more accurate than at intermediate points [20]. It can be seen in Figure 3(a) that for two out of three cases, the error with the adaptive grid is mostly smaller than the error with a uniform grid with the same number of points.…”

Section: Accuracy and Efficiency In Practicementioning

confidence: 51%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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The Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. AbstractThis paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples.

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Section: Accuracy and Efficiency In Practicementioning

confidence: 51%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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“…The Euler-Lagrange equation requires four boundary conditions in order to have a unique solution. For details, see [15]. In particular, for the steady state power-law Euler-Bernoulli beam, we have…”

Section: The Ritz Variational Principlementioning

confidence: 99%

Analytic and finite element solutions of the power-law Euler–Bernoulli beams

Wei

Liu

2012

Finite Elements in Analysis and Design

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In this paper, we use Hermite cubic finite elements to approximate the solutions of a nonlinear Euler-Bernoulli beam equation. The equation is derived from Hollomon's generalized Hooke's law for work hardening materials with the assumptions of the Euler-Bernoulli beam theory. The Ritz-Galerkin finite element procedure is used to form a finite dimensional nonlinear program problem, and a nonlinear conjugate gradient scheme is implemented to find the minimizer of the Lagrangian. Convergence of the finite element approximations is analyzed and some error estimates are presented. A Matlab finite element code is developed to provide numerical solutions to the beam equation. Some analytic solutions are derived to validate the numerical solutions. To our knowledge, the numerical solutions as well as the analytic solutions are not available in the literature.

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“…where ϕj and ψi are the trial and test functions from the trial and test spaces respectively [3,13]. S is typically sparse due to the local support of the chosen basis functions.…”

Section: Introductionmentioning

confidence: 99%

Symbolic integration at compile time in finite element methods

Rupp

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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In most existing software packages for the finite element method it is not possible to supply the weak formulation of the problem of interest in a compact form, which was in the early days of programming due to the low abstraction capabilities of available programming languages. With the advent of pure object-oriented programming, abstraction was long said to be achievable only in trade-off with run time efficiency. In this work we show that it is possible to obtain both a high level of abstraction and good run time efficiency by the use of template metaprogramming in C++. We focus on a mathematical expressions engine, by which element matrices are computed during compile time and by which the weak formulation can be specified in a single line of code. A comparison of system matrix assembly times of existing finite element software shows that the template metaprogramming approach is up to an order of magnitude faster than traditional software designs.

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Finite Element Solution of Boundary Value Problems

Cited by 132 publications

References 0 publications

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Analytic and finite element solutions of the power-law Euler–Bernoulli beams

Symbolic integration at compile time in finite element methods

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