Low-complexity computation of plate eigenmodes with Vekua approximations and the Method of Particular Solutions

Authors: Gilles Chardon, Laurent Daudet

arXiv: 1301.0908v2 - DOI (math.NA)
Submitted to arXiv on 05 Jan. 2013
Abstract

This paper extends the Method of Particular Solutions (MPS) to the computation of eigenfrequencies and eigenmodes of plates. Specific approximation schemes are developed, with plane waves (MPS-PW) or Fourier-Bessel functions (MPS-FB). This framework also requires a suitable formulation of the boundary conditions. Numerical tests, on two plates with various boundary conditions, demonstrate that the proposed approach provides competitive results with standard numerical schemes such as the Finite Element Method, at reduced complexity, and with large flexibility in the implementation choices.

The papers around it, by date and closeness

Higher is closer to this paper; left is older. Lines join papers close to each other: groups of lines are sub-topics. Select a point to see why it is there. Back to the summary of this paper

Closest papers Prior work it builds on Papers that cite it

Papers around this one

Ranked by how close their title and abstract are to this paper’s. Select some to read them together.

Closest papers

  1. Related
    A robust solution strategy for the Cahn-Larché equations
    Erlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Numerical Analysis · AI summary available
    Abstract

    In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment…

  2. Related
    The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
    Matthew J. Colbrook · 2022 · Numerical Analysis · AI summary available
    Abstract

    Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition ($\texttt{mpEDMD}$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. $\texttt{mpEDMD}$ is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman op…

  3. Related
    Compatible finite element methods for geophysical fluid dynamics
    Colin J. Cotter · 2023 · Numerical Analysis · AI summary available
    Abstract

    This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz d…

  4. Related
    Thirty years of the finite volume method for solid mechanics
    Philip Cardiff, Ismet Demirdžić · 2018 · Numerical Analysis · AI summary available
    Abstract

    Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finish…

  5. Related
    Smooth multi-patch scaled boundary isogeometric analysis for Kirchhoff-Love shells
    Mathias Reichle, Jeremias Arf, Bernd Simeon et al. · 2023 · Numerical Analysis · AI summary available
    Abstract

    In this work, a linear Kirchhoff-Love shell formulation in the framework of scaled boundary isogeometric analysis is presented that aims to provide a simple approach to trimming for NURBS-based shell analysis. To obtain a global C1-regular test function space for the shell discretization, an inter-patch coupling is applied with adjusted basis functions in the vicinity of the scaling center to ensure the approximation ability. Doing so, the scaled boundary geometries are related to the concept of analysis-suitable G1 parametrizations. This yields a coupling of patch boundaries in a strong sense that is restricted to G1-smooth surfaces. The proposed approach is advantageous to trimmed geometri…

  6. Related
    A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem
    Ruo Li, Pingbing Ming, Zhiyuan Sun et al. · 2017 · Numerical Analysis · Not summarized yet
    Abstract

    We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.

  7. Related
    A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation
    Ritukesh Bharali, Somdatta Goswami, Cosmin Anitescu et al. · 2021 · Numerical Analysis · AI summary available
    Abstract

    The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adop…

  8. Related
    A Computationally Efficient Vectorized Implementation of Localizing Gradient Damage Method in MATLAB
    Subrato Sarkar · 2023 · Numerical Analysis · AI summary available
    Abstract

    In this work, a recently developed fracture modeling method called localizing gradient damage method (LGDM) is implemented in MATLAB. MATLAB is well-known in the computational research community for its simple and easy-to-learn coding interface. As a result, MATLAB is generally preferred for the initial development (prototyping) of computational models by researchers. However, MATLAB-developed codes are seldom used for large-scale simulations (after initial development is complete) due to their computational inefficiency. Hence, a computationally efficient implementation of LGDM using MATLAB vectorization is presented in this work. The choice of LGDM (as the fracture modeling method) is base…

How are these papers found?

We compare the title and abstract of this paper with those of every paper summarized on this site, and with the papers Semantic Scholar recommends, those it cites and those citing it (for a much-cited paper, its most recent citations and, from OpenAlex, the most cited ones). Among the prior work and the follow-ups, the papers that use each other in depth, a method it uses and the much-cited follow-ups come a little higher. The levels (very close, close, related) come from the similarity of the texts, not from a judgment of quality; * marks a paper without an abstract, compared by its title. The sentence under a paper is written by an AI from the two abstracts: check it in the papers. A paper marked not relevant by three readers leaves the list.

For now only among the papers summarized on this site: search again to add the prior work, the follow-ups and Semantic Scholar’s recommendations.