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Recent Advances in Finite Element Methods with Applications
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Recent Advances in Finite Element Methods with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 15 September 2024 | Viewed by 7853

Special Issue Editor

Dr. Shuo Zhang

E-Mail Website
Guest Editor
1. State Key Laboratory of Scientific and Engineering Computing (LSEC), Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
2. University of Chinese Academy of Sciences, Beijing 100049, China
Interests: numerical analysis; finite element method; structure preservation; multilevel method

Special Issue Information

Dear Colleagues,

The finite element method is an important tool used in applied sciences. In close association with computational mechanics, it has been increasingly applied across various fields, such as engineering, material sciences, environmental sciences, medicine, biology, as well as physics and chemistry, and so forth. The finite element method also motivates extensive research on mathematics, providing specific structures for the firm theoretical foundation.

This Special Issue, entitled “Recent Advances in Finite Element Methods with Applications”, aims to collect recent advances in the construction, theoretical analysis, implementation, and application of finite element methods. We invite investigators to contribute high-quality original research articles as well as review articles on recent advances in the following methods:

  • finite element algorithms and mathematical theories for both classical and new model problems;
  • applications of the method for real world problems, either on a specific problem or about the trend of a whole area, where finite element methods are used as research tools or as conceptual foundations;
  • developments and principles of finite element software packages and platforms, as well as new techniques for a mid-way step, such as mesh generation.

Potential topics include, but are not limited to, Navier–Stokes equations, Magnetohydrodynamic equations, Boussinesq equations, Einstein equation, large deformation elasticity, computational biomechanics and biomathematics, medical engineering, mathematical theories of finite element methods, the interplay of finite element methods and machine learning, and so forth. The Special Issue is open to all kinds of finite element methods, and to advanced implementation approaches of the methods.

Dr. Shuo Zhang
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Finite element methods
  • Mixed finite element methods
  • Spectral element method
  • Discontinuous Galerkin methods
  • Grid methods
  • Meshfree methods
  • Loubignac iteration
  • Virtual element method
  • Hpk-FEM
  • Navier–Stokes equations
  • Magnetohydrodynamic equations
  • Boussinesq equations
  • Large deformation elasticity
  • Algorithms of FEM
  • Applications of FEM
  • Machine learning
  • Implementation techniques of FEM
  • Computational physics, chemistry and mechanics
  • Computational biomechanics and biomathematics
  • Material modeling
  • Computational applied sciences

Published Papers (6 papers)

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Research

18 pages, 388 KiB  
Article
Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons
by Todd Arbogast and Chuning Wang
Mathematics 2023, 11(22), 4663; https://doi.org/10.3390/math11224663 - 16 Nov 2023
Viewed by 689
Abstract
New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order [...] Read more.
New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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29 pages, 6181 KiB  
Article
Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain
by Xudong Jiang, Wei Zhang, Xiaoyan Teng and Xiangyang Chen
Mathematics 2023, 11(16), 3488; https://doi.org/10.3390/math11163488 - 12 Aug 2023
Viewed by 880
Abstract
This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization [...] Read more.
This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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18 pages, 7828 KiB  
Article
Simulation of Electromagnetic Forming Process and Optimization of Geometric Parameters of Perforated Al Sheet Using RSM
by Nilesh Satonkar and Venkatachalam Gopalan
Mathematics 2023, 11(9), 1983; https://doi.org/10.3390/math11091983 - 22 Apr 2023
Cited by 2 | Viewed by 1368
Abstract
Electromagnetic forming (EMF) is a kind of high-speed forming technology that can be useful for materials like aluminum. EMF helps to overcome the limitations of traditional forming. Due to this ability, the use of EMF in automotive applications has risen in recent years. [...] Read more.
Electromagnetic forming (EMF) is a kind of high-speed forming technology that can be useful for materials like aluminum. EMF helps to overcome the limitations of traditional forming. Due to this ability, the use of EMF in automotive applications has risen in recent years. The application of finite element software packages such as ANSYS 22 gives numerical modelling capabilities to simulate the EMF process and to design the forming process. Hence, the aim of this research work is to build and study the three-dimensional finite element model for the electromagnetic forming process and analyze the geometric parameters influencing the deformation of the perforated sheet with a design of experiments (DOE) approach. The finite element simulation is used in two stages. In the first stage, the electromagnetic force or Lorentz force striking the workpiece (i.e., Al sheet) is predicted using the ANSYS 22 Emag module. In the second stage, the predicted Lorentz force is then applied on an Al sheet to calculate the sheet deformation. The deformation of the sheet is predicted for different combinations of the geometric parameters of the sheet, such as open area percentage, ligament ratio (LR) and size of the hole, using ANSYS 22 Structural. In the DOE, response surface methodology (RSM) is used by considering the geometric parameters of perforated sheet such as open area percentage, ligament ratio (LR) and size of the hole. To minimize the number of experiments, an RSM model named central composite design (CCD) is employed. Further, the optimization study finds that the maximum deformation 0.0435 mm is calculated for the optimized combination of 25% open area, 0.14 LR and 4 mm hole size. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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25 pages, 13573 KiB  
Article
A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography
by Md Mamunur Rasid, Masato Kimura, Md Masum Murshed, Erny Rahayu Wijayanti and Hirofumi Notsu
Mathematics 2023, 11(7), 1633; https://doi.org/10.3390/math11071633 - 28 Mar 2023
Viewed by 1441
Abstract
A two-step Lagrange–Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC) is presented. First, we show the experimental order of convergence to see the second-order accuracy in time realized by the two-step methods for conservative and non-conservative material derivatives [...] Read more.
A two-step Lagrange–Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC) is presented. First, we show the experimental order of convergence to see the second-order accuracy in time realized by the two-step methods for conservative and non-conservative material derivatives along the trajectory of fluid particles. Second, we observe the effect of the TBC in a simple domain, and the artificial reflection is removed significantly when the wave touches the TBC. Third, we apply the scheme to a practical domain with islands, namely, the Bay of Bengal region, and observe the effect of the TBC again for the practical domain; the artificial reflections are removed significantly from the transmission boundaries on open sea boundaries. We also study the effect of a position of an open sea boundary with the TBC and reveal that it is sufficiently small to neglect. The numerical results in this study show that the scheme has the following properties: (i) the same advantages of Lagrange–Galerkin methods (the CFL-free robustness for convection-dominated problems and the symmetry of the matrices for the system of linear equations); (ii) second-order accuracy in time by the two-step methods; (iii) mass preservation of the function for the water level from the reference height (until the contact with the transmission boundaries of the wave); and (iv) no significant artificial reflection from the transmission boundaries. The numerical results by the scheme presented in this paper are for the flat bottom topography of the domain. In the next part of this work, Part II, the scheme will be applied to rapidly varying bottom surfaces and a real bottom topography of the Bay of Bengal region. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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19 pages, 9684 KiB  
Article
Generalized Thermoelastic Interaction in Orthotropic Media under Variable Thermal Conductivity Using the Finite Element Method
by Aatef Hobiny and Ibrahim Abbas
Mathematics 2023, 11(4), 955; https://doi.org/10.3390/math11040955 - 13 Feb 2023
Viewed by 1091
Abstract
This article addresses a thermoelastic problem under varying thermal conductivity with and without Kirchhoff’s transforms. The temperature increment, displacement, and thermal stresses in an orthotropic material with spherical cavities are studied. The inner surface of the hole is constrained and heated by thermal [...] Read more.
This article addresses a thermoelastic problem under varying thermal conductivity with and without Kirchhoff’s transforms. The temperature increment, displacement, and thermal stresses in an orthotropic material with spherical cavities are studied. The inner surface of the hole is constrained and heated by thermal shock. The numerical solutions are derived using the finite element technique in the setting of the generalized thermoelasticity model with one thermal delay time. The thermal conductivity of the material is supposed to be temperature-dependent without Kirchhoff’s transformation. Due to the difficulty of nonlinear formulations, the finite element approach is used to solve the problem without using Kirchhoff’s transformation. The solution is determined using the Laplace transform and the eigenvalues technique when employing Kirchhoff’s transformation in a linear example. Variable thermal conductivity is addressed and compared with and without Kirchhoff’s transformation. The numerical result for the investigated fields is graphically represented. According to the numerical analysis results, the varying thermal conductivity provides a limited speed for the propagations of both mechanical and thermal waves. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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32 pages, 748 KiB  
Article
Approximate Solution of Two Dimensional Disc-like Systems by One Dimensional Reduction: An Approach through the Green Function Formalism Using the Finite Elements Method
by Alejandro Ferrero and Juan Pablo Mallarino
Mathematics 2023, 11(1), 197; https://doi.org/10.3390/math11010197 - 30 Dec 2022
Viewed by 1444
Abstract
We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure [...] Read more.
We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results. Full article
(This article belongs to the Special Issue Recent Advances in Finite Element Methods with Applications)
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