Статті в журналах: "FEM discretization"

1

Dryja, M., and M. Sarkis. "Additive Average Schwarz Methods for Discretization of Elliptic Problems with Highly Discontinuous Coefficients." Computational Methods in Applied Mathematics 10, no. 2 (2010): 164–76. http://dx.doi.org/10.2478/cmam-2010-0009.

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AbstractA second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the second discretization. It has been established that the condition number of the preconditioned systems does not depend on the jumps of the coefficients across the substructure boundaries and outside of a thin layer along the substructure boundaries. The algorithms are very well suited for parallel computations.

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2

Martello, Giulia. "Discretization Analysis in FEM Models." MATEC Web of Conferences 53 (2016): 01063. http://dx.doi.org/10.1051/matecconf/20165301063.

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3

Lahtinen, Valtteri, and Antti Stenvall. "A category theoretical interpretation of discretization in Galerkin finite element method." Mathematische Zeitschrift 296, no. 3-4 (January 29, 2020): 1271–85. http://dx.doi.org/10.1007/s00209-020-02456-1.

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Abstract The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.

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4

MARAZZINA, DANIELE, OLEG REICHMANN, and CHRISTOPH SCHWAB. "hp-DGFEM FOR KOLMOGOROV–FOKKER–PLANCK EQUATIONS OF MULTIVARIATE LÉVY PROCESSES." Mathematical Models and Methods in Applied Sciences 22, no. 01 (January 2012): 1150005. http://dx.doi.org/10.1142/s0218202512005897.

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We analyze the discretization of nonlocal degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes. Such equations arise in option pricing problems when the stochastic dynamics of the markets is modeled by Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes, in particular the discontinuous Galerkin Finite Element Methods (DG-FEM). In the DG-FEM, a new regularization of hypersingular integrals in the Dirichlet form of the pure jump part of infinite variation processes is proposed, allowing in particular a stable DG discretization of hypersingular integral operators. Robustness of the stabilized discretization with respect to various degeneracies in the characteristic triple of the stochastic process is proved. We provide in particular an hp-error analysis of the DG-FEM. Numerical experiments for model equations confirm the theoretical results.

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5

Ovchinnikov, George V., Denis Zorin, and Ivan V. Oseledets. "Robust regularization of topology optimization problems with a posteriori error estimators." Russian Journal of Numerical Analysis and Mathematical Modelling 34, no. 1 (February 25, 2019): 57–69. http://dx.doi.org/10.1515/rnam-2019-0005.

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Abstract Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material). Using a finite elements discretization (FEM) of the PDE and functional we obtain an integer programming problem. Due to approximation error of the FEM discretization, optimization problem becomes mesh-depended and possess false, physically inadequate optimums, while functional value heavily depends on the fineness of discretization scheme used to compute it. To alleviate this problem, we propose regularization of given functional by error estimate of the FEM discretization. This regularization provides robustness of solutions and improves obtained functional values as well. While the idea is broadly applicable, in this paper we apply our method to the heat conduction optimization. Problems of this type are of practical importance in design of heat conduction channels, heat sinks and other types of heat guides.

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6

Schedensack, Mira. "A New Generalization of the P1 Non-Conforming FEM to Higher Polynomial Degrees." Computational Methods in Applied Mathematics 17, no. 1 (January 1, 2017): 161–85. http://dx.doi.org/10.1515/cmam-2016-0031.

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AbstractThis paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

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7

Devaud, Denis. "Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2." IMA Journal of Numerical Analysis 40, no. 4 (October 16, 2019): 2717–45. http://dx.doi.org/10.1093/imanum/drz036.

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Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.

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8

Yao, Lingyun, Wanyi Tian, and Fei Wu. "An Optimized Generalized Integration Rules for Error Reduction of Acoustic Finite Element Model." International Journal of Computational Methods 15, no. 07 (October 12, 2018): 1850062. http://dx.doi.org/10.1142/s0219876218500627.

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In the finite element method (FEM), the accuracy in acoustic problems will deteriorate with the increasing frequency due to the “dispersion effect”. In order to minimize discretization error, a novel optimized generalized integration rules (OGIR) is introduced into FEM for the reduction of discretization error. In the present work, the adaptive genetic algorithm (AGA) is implemented to sight the optimized location of integration points. Firstly, the generalized integration rules (GIR) is used to parameterize the Gauss point location, then the relationship between the location parameterize of the integration points and discretization error is derived in detail, and the optimized location of the integration points is found through the optimization procedure, and then the OGIR–FEM is finally proposed to solve the acoustic problem. It also can be directly used to solve the optional acoustic problem, including the damped problems. Numerical example involving distorted meshes indicates that present OGIR–FEM has a superior error reducing performance in comparison with the other error reducing finite elements. These researches indicate that the proposed method can be more widely applied to solving practical acoustic problems with more accurate solutions.

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9

Zhao, Jingjun, Jingyu Xiao, and Yang Xu. "Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/857205.

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A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.

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10

Xu, Haochen. "Analyzing heat transfer in Axial Flux Permanent Magnet electrical machines: A literature review on the discretization methods-FVM and FDM." Theoretical and Natural Science 11, no. 1 (November 17, 2023): 223–30. http://dx.doi.org/10.54254/2753-8818/11/20230412.

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Axial Flux Permanent Magnet (AFPM) machines have gained significant attention due to their high power density, efficiency, and compact design. However, effective heat transfer analysis is critical for optimizing their performance and reliability. This paper presents a comprehensive literature review on the application of discretization methods, specifically the Finite Volume Method (FVM) and Finite Difference Method (FDM), in the thermal analysis of AFPM machines. The fundamentals of FVM and FDM are briefly explained, followed by an exploration of their applications in AFPM machine thermal analysis. The advantages and limitations of using these methods are discussed, and a comparison between FVM and FDM is provided. Advanced discretization techniques, such as the Finite Element Method (FEM), and coupled thermal-electromagnetic analyses are also examined. The paper highlights studies that have utilized FVM or FDM in developing optimized designs and effective thermal management strategies for AFPM machines. Lastly, potential future research directions are identified, including the development of more efficient discretization methods, the incorporation of advanced materials, and the investigation of novel cooling techniques. This review offers valuable insights into the current state of research and potential future directions in the thermal analysis of AFPM machines using discretization methods.

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11

Li, Long-yuan, and Peter Bettess. "Adaptive Finite Element Methods: A Review." Applied Mechanics Reviews 50, no. 10 (October 1, 1997): 581–91. http://dx.doi.org/10.1115/1.3101670.

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The adaptive finite element method (FEM) was developed in the early 1980s. The basic concept of adaptivity developed in the FEM is that, when a physical problem is analyzed using finite elements, there exist some discretization errors caused owing to the use of the finite element model. These errors are calculated in order to assess the accuracy of the solution obtained. If the errors are large, then the finite element model is refined through reducing the size of elements or increasing the order of interpolation functions. The new model is re-analyzed and the errors in the new model are recalculated. This procedure is continued until the calculated errors fall below the specified permissible values. The key features in the adaptive FEM are the estimation of discretization errors and the refinement of finite element models. This paper presents a brief review of the methods for error estimates and adaptive refinement processes applied to finite element calculations. The basic theories and principles of estimating finite element discretization errors and refining finite element models are presented. This review article contains 131 references.

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12

Zhao, Jingjun, Jingyu Xiao, and Yang Xu. "A Finite Element Method for the Multiterm Time-Space Riesz Fractional Advection-Diffusion Equations in Finite Domain." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/868035.

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We present an effective finite element method (FEM) for the multiterm time-space Riesz fractional advection-diffusion equations (MT-TS-RFADEs). We obtain the weak formulation of MT-TS-RFADEs and prove the existence and uniqueness of weak solution by the Lax-Milgram theorem. For multiterm time discretization, we use the Diethelm fractional backward finite difference method based on quadrature. For spatial discretization, we show the details of an FEM for such MT-TS-RFADEs. Then, stability and convergence of such numerical method are proved, and some numerical examples are given to match well with the main conclusions.

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13

Pal, Mahendra Kumar, M. L. L. Wijerathne, and Muneo Hori. "Numerical Modeling of Brittle Cracks Using Higher Order Particle Discretization Scheme–FEM." International Journal of Computational Methods 16, no. 04 (May 13, 2019): 1843006. http://dx.doi.org/10.1142/s0219876218430065.

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Higher order extension of Particle Discretization Scheme (HO-PDS), its implementation in FEM framework (HO-PDS-FEM) and applications in efficiently simulating cracks are presented in this paper. PDS is an approximation scheme which uses a conjugate domain tessellation pair like Voronoi and Delaunay in approximating a function and its derivatives. In approximating a function (or derivatives), HO-PDS first produces local polynomial approximations for the target function (or derivatives) within each element of respective tessellation. The approximations over the whole domain are then obtained by taking the union of those respective local approximations. These approximations are inherently discontinuous along the boundaries of the respective tessellation elements since the support of the local approximations is confined to the domain of respective tessellation elements and no continuity conditions are enforced. HO-PDS-FEM utilizes these inherent discontinuities in function approximation to efficiently model discontinuities such as cracks. Higher order PDS is implemented in FEM framework to solve boundary value problem of elastic solids, including mode-I crack problems. With several benchmark problems, it is shown that HO-PDS-FEM has higher expected accuracy and convergence rate. J-integral around a mode-I crack tip is calculated to demonstrate the improvement in the accuracy of the crack tip stress field. Further, it is shown that HO-PDS-FEM significantly improves the traction along the crack surfaces, compared to the zeroth-order PDS-FEM [Hori, M., Oguni, K. and Sakaguchi, H. [2005] “Proposal of FEM implemented with particle discretization scheme for analysis of failure phenomena,” J. Mech. Phys. Solids 53, 681–703].

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14

Korga, Sylwester, Anna Makarewicz, and Klaudiusz Lenik. "METHODS OF DISCRETIZATION OBJECTS CONTINUUM IMPLEMENTED IN FEM PREPROCESSORS." Advances in Science and Technology Research Journal 9, no. 28 (2015): 130–33. http://dx.doi.org/10.12913/22998624/60800.

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15

El Moutea, Omar, Nadia Nakbi, Abdeslam El Akkad, Ahmed Elkhalfi, Lahcen El Ouadefli, Sorin Vlase, and Maria Luminita Scutaru. "A Mixed Finite Element Approximation for Time-Dependent Navier–Stokes Equations with a General Boundary Condition." Symmetry 15, no. 11 (November 8, 2023): 2031. http://dx.doi.org/10.3390/sym15112031.

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In this paper, we present a numerical scheme for addressing the unsteady asymmetric flows governed by the incompressible Navier–Stokes equations under a general boundary condition. We utilized the Finite Element Method (FEM) for spatial discretization and the fully implicit Euler scheme for time discretization. In addition to the theoretical analysis of the error in our numerical scheme, we introduced two types of a posteriori error indicators: one for time discretization and another for spatial discretization, aimed at effectively controlling the error. We established the equivalence between these estimators and the actual error. Furthermore, we conducted numerical simulations in two dimensions to assess the accuracy and effectiveness of our scheme.

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16

Liu, Xiang, Guo-ping Cai, Fu-jun Peng, Hua Zhang, and Liang-liang Lv. "Nonlinear vibration analysis of a membrane based on large deflection theory." Journal of Vibration and Control 24, no. 12 (January 9, 2017): 2418–29. http://dx.doi.org/10.1177/1077546316687924.

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This paper investigates nonlinear vibration of a simply supported rectangular membrane based on large deflection theory. Dynamic stress caused by transverse displacement of the membrane is considered in modeling the membrane. The assumed mode method and the nonlinear finite element method (FEM) are both used as discretization methods for the membrane. In the assumed mode method, an approximate analytical formula of the natural frequency is derived. In the nonlinear FEM, a three-node triangular membrane element is proposed. The difference between the membrane’s dynamical characteristics obtained by these two discretization methods is revealed. Simulation results indicate that natural frequency of the membrane will rise along with the increasing of the vibration amplitude of the membrane, and the natural frequency obtained by the nonlinear FEM is larger than that obtained by the assumed mode method. When the membrane vibration is small, the assumed mode method may achieve a reasonable result, but it may lead to a big error when the membrane vibration is large.

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Klochkov, Yuriy V., Valeria A. Pshenichkina, Anatoliy P. Nikolaev, Olga V. Vakhnina, and Mikhail Yu Klochkov. "Quadrilateral element in mixed FEM for analysis of thin shells of revolution." Structural Mechanics of Engineering Constructions and Buildings 19, no. 1 (March 30, 2023): 64–72. http://dx.doi.org/10.22363/1815-5235-2023-19-1-64-72.

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The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.

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18

Mariano, Valeria, Jorge A. Tobon Vasquez, and Francesca Vipiana. "A Novel Discretization Procedure in the CSI-FEM Algorithm for Brain Stroke Microwave Imaging." Sensors 23, no. 1 (December 20, 2022): 11. http://dx.doi.org/10.3390/s23010011.

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In this work, the contrast source inversion method is combined with a finite element method to solve microwave imaging problems. The paper’s major contribution is the development of a novel contrast source variable discretization that leads to simplify the algorithm implementation and, at the same time, to improve the accuracy of the discretized quantities. Moreover, the imaging problem is recreated in a synthetic environment, where the antennas, and their corresponding coaxial port, are modeled. The implemented algorithm is applied to reconstruct the tissues’ dielectric properties inside the head for brain stroke microwave imaging. The proposed implementation is compared with the standard one to evaluate the impact of the variables’ discretization on the algorithm’s accuracy. Furthermore, the paper shows the obtained performances with the proposed and the standard implementations of the contrast source inversion method in the same realistic 3D scenario. The exploited numerical example shows that the proposed discretization can reach a better focus on the stroke region in comparison with the standard one. However, the variation is within a limited range of permittivity values, which is reflected in similar averages.

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19

Beuchler, Sven, and Martin Purrucker. "Schwarz Type Solvers for -FEM Discretizations of Mixed Problems." Computational Methods in Applied Mathematics 12, no. 4 (2012): 369–90. http://dx.doi.org/10.2478/cmam-2012-0030.

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AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

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20

Huszty, Csaba, and Ferenc Izsák. "Symplectic time-domain finite element method (STD-FEM) for room acoustic modeling." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 268, no. 5 (November 30, 2023): 3089–99. http://dx.doi.org/10.3397/in_2023_0447.

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A new, extended symplectic time-domain finite element method (STD-FEM) is proposed for room acoustic modeling. As a mathematical model for sound propagation and reflection, the classical time-domain wave equations are used, which can be extended with air absorption. Frequency-dependent locally reactive boundary conditions are also introduced to the model. Spatially, a third-order tensor product type spectral element discretization is applied, which allows us to use explicit time steps. For this purpose, a partitioned Runge-Kutta method is employed, which is an extension of a third-order symplectic time-discretization. The method is validated on a number of examples and the performance and stability benefits are presented. We present the implementation of the method in the soundy.ai application.

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21

Koleva, Miglena. "FINITE ELEMENT SOLUTION OF BOUNDARY VALUE PROBLEMS WITH NONLOCAL JUMP CONDITIONS." Mathematical Modelling and Analysis 13, no. 3 (September 30, 2008): 383–400. http://dx.doi.org/10.3846/1392-6292.2008.13.383-400.

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We consider stationary linear problems on non‐connected layers with distinct material properties. Well posedness and the maximum principle (MP) for the differential problems are proved. A version of the finite element method (FEM) is used for discretization of the continuous problems. Also, the MP and convergence for the discrete solutions are established. An efficient algorithm for solution of the FEM algebraic equations is proposed. Numerical experiments for linear and nonlinear problems are discussed.

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22

Liu, G. R. "On Partitions of Unity Property of Nodal Shape Functions: Rigid-Body-Movement Reproduction and Mass Conservation." International Journal of Computational Methods 13, no. 02 (March 2016): 1640003. http://dx.doi.org/10.1142/s021987621640003x.

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This paper discusses the Partitions of Unity (PU) property that is one of the most important properties of nodal shape functions used in various numerical methods via discretization, including element-based and/or meshfree methods, such as FEM, S-FEM, S-PIM, EFG, XFEM, etc. The significance of the PU property and the possible consequences of using shape functions that do not possess the PU property in a numerical method are examined in theory. It proves that the PU property is a necessary (not sufficient in general) condition to enable the basic feature of rigid-body-movement production for static problems, and it is both necessary and sufficient condition mass conservation for dynamic problems for solids. This paper offers a fundamental insight into the lack of essential solution properties when formulating a computational method based on discretization techniques with shape functions that do not possess the PU property.

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Gungor, Arif Can, Marzena Olszewska-Placha, Malgorzata Celuch, Jasmin Smajic, and Juerg Leuthold. "Advanced Modelling Techniques for Resonator Based Dielectric and Semiconductor Materials Characterization." Applied Sciences 10, no. 23 (November 29, 2020): 8533. http://dx.doi.org/10.3390/app10238533.

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This article reports recent developments in modelling based on Finite Difference Time Domain (FDTD) and Finite Element Method (FEM) for dielectric resonator material measurement setups. In contrast to the methods of the dielectric resonator design, where analytical expansion into Bessel functions is used to solve the Maxwell equations, here the analytical information is used only to ensure the fixed angular variation of the fields, while in the longitudinal and radial direction space discretization is applied, that reduced the problem to 2D. Moreover, when the discretization is performed in time domain, full-wave electromagnetic solvers can be directly coupled to semiconductor drift-diffusion solvers to better understand and predict the behavior of the resonator with semiconductor-based samples. Herein, FDTD and frequency domain FEM approaches are applied to the modelling of dielectric samples and validated against the measurements within the 0.3% margin dictated by the IEC norm. Then a coupled in-house developed multiphysics time-domain FEM solver is employed in order to take the local conductivity changes under electromagnetic illumination into account. New methodologies are thereby demonstrated that open the way to new applications of the dielectric resonator measurements.

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Chukwuyem, Nwankwo Jude, Njoseh Ignatius Nkonyeasua, and Joshua Sarduana Apanapudor. "Runge-Kutta Finite Element Method for the Fractional Stochastic Wave Equation." Journal of Advances in Mathematics and Computer Science 39, no. 12 (November 30, 2024): 70–83. https://doi.org/10.9734/jamcs/2024/v39i121950.

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This paper presents the development and application of the Runge-Kutta Finite Element Method (RK-FEM) to solve fractional stochastic wave equations. Fractional differential equations (FDEs) play a significant role in modelling complex systems with memory and hereditary properties, while the inclusion of stochastic components accounts for randomness inherent in physical systems. The fractional stochastic wave equation represents a natural extension of classical wave equations, incorporating both fractional time derivatives and stochastic processes to model phenomena such as anomalous diffusion and noise-driven wave propagation. We propose a hybrid numerical scheme that combines the high accuracy of the Runge-Kutta Method or temporal discretization with the flexibility of the Finite Element Method (FEM) for spatial discretization. The Caputo fractional derivative is used to describe the time-fractional component of the equation. A white noise-driven stochastic term is incorporated into the system to account for randomness. We analyze the stability and convergence properties of the RK-FEM scheme and demonstrate its effectiveness through numerical simulations. The results illustrate that the proposed method provides accurate and stable solutions for fractional stochastic wave equations, making it a robust tool for investigating wave phenomena in complex and uncertain environments.

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25

Yang, Yidu. "Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems." Journal of Computational Mathematics 27, no. 6 (June 2009): 748–63. http://dx.doi.org/10.4208//jcm.2009.09-m2876.

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26

Solin, P., J. Cerveny, L. Dubcova, and D. Andrs. "Monolithic discretization of linear thermoelasticity problems via adaptive multimesh hp-FEM." Journal of Computational and Applied Mathematics 234, no. 7 (August 2010): 2350–57. http://dx.doi.org/10.1016/j.cam.2009.08.092.

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27

Zdechlik, Robert, and Agnieszka Kałuża. "The FEM model of groundwater circulation in the vicinity of the Świniarsko intake, near Nowy Sącz (Poland)." Geologos 25, no. 3 (December 1, 2019): 255–62. http://dx.doi.org/10.2478/logos-2019-0028.

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Abstract Modern hydrogeological research uses numerical modelling, which is most often based on the finite difference method (FDM) or finite element method (FEM). The present paper discusses an example of application of the less frequently used FEM for simulating groundwater circulation in the vicinity of the intake at Świniarsko near Nowy Sącz. The research area is bordered by rivers and watersheds, and within it, two well-connected aquifers occur (Quaternary gravelly-sandy sediments and Paleogene cracked flysch rocks). The area was discretized using a Triangle generator, taking into account assumptions about the nature and density of the mesh. Rivers, wells, an irrigation ditch and infiltration of precipitation were projected onto boundary conditions. Conditions of groundwater circulation in the aquifer have been assessed based on a calibrated model, using water balance and a groundwater level contour map with flow path lines. Application of the program based on FEM, using smooth local densification of the discretization mesh, has allowed for precise mapping of the location of objects that significantly shape water circulation.

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Kahla, Nabil Ben, Saeed AlQadhi, and Mohd Ahmed. "Radial Point Interpolation-Based Error Recovery Estimates for Finite Element Solutions of Incompressible Elastic Problems." Applied Sciences 13, no. 4 (February 12, 2023): 2366. http://dx.doi.org/10.3390/app13042366.

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Error estimation and adaptive applications help to control the discretization errors in finite element analysis. The study implements the radial point interpolation (RPI)-based error-recovery approaches in finite element analysis. The displacement/pressure-based mixed approach is used in finite element formulation. The RPI approach considers the radial basis functions (RBF) and polynomials basis functions together to interpolate the finite element solutions, i.e., displacement over influence zones to recover the solution errors. The energy norm is used to represent global and local errors. The reliability and effectiveness of RPI-based error-recovery approaches are assessed by adaptive analysis of incompressibility elastic problems including the problem with singularity. The quadrilateral meshes are used for discretization of problem domains. For adaptive improvement of mesh, the square of error equally distributed technique is employed. The computational outcome for solution errors, i.e., error distribution and convergence rate, are obtained for RPI technique-based error-recovery approach employing different radial basis functions (multi quadratic, thin-plate splint), RBF shape parameters, different shapes of influence zones (circular, rectangular) and conventional patches. The error convergence in the original FEM solution, in FEM solution considering influence-zone-based RPI recovery with MQ RBF, conventional patch-based RPI recovery with MQ RBF and conventional patch LS-based error recovery are found as (0.97772, 2.03291, 1.97929 and 1.6740), respectively, for four-node quadrilateral discretization of problem, while for nine-node quadrilateral discretization, the error convergence is (1.99607, 3.53087, 4.26621 and 2.54955), respectively. The study concludes that the adaptive analysis, using error-recovery estimates-based RPI approach, provides results with excellent accuracy and reliability.

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Chen, Hao, and Li Li Xie. "Extension to Elasto-Plastic Version of a Fracture Mechanics Method." Applied Mechanics and Materials 703 (December 2014): 376–80. http://dx.doi.org/10.4028/www.scientific.net/amm.703.376.

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This paper develops a three dimensional elastic fracture analysis method, PDS-FEM (Particle Discretization Scheme Finite Element Method), to its elasto-plastic version. The Newton-Raphson iteration method is adopted for solving material nonlinearity, and the conjugate gradient method is applied to solve the linear equations of FEM. In order to apply the fracture analysis method to the engineering scale analysis, CPU based parallel computing technology is applied, and the computation speed is highly advanced. In this trial test, a simple stress based failure criterion is employed for the failure analysis of a cantilever steel beam. The numerical results without fracture match well with the commercial FEM software, ANSYS’s, which verifies the accuracy of the developed platform.

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Nadal, E., J. J. Ródenas, J. Albelda, M. Tur, J. E. Tarancón, and F. J. Fuenmayor. "Efficient Finite Element Methodology Based on Cartesian Grids: Application to Structural Shape Optimization." Abstract and Applied Analysis 2013 (2013): 1–19. http://dx.doi.org/10.1155/2013/953786.

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This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

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31

Reddy, Gujji Murali Mohan, Alan B. Seitenfuss, Débora de Oliveira Medeiros, Luca Meacci, Milton Assunção, and Michael Vynnycky. "A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D." Algorithms 13, no. 10 (September 24, 2020): 242. http://dx.doi.org/10.3390/a13100242.

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Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

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Liu, Chao, and Robert G. Kelly. "A Review of the Application of Finite Element Method (FEM) to Localized Corrosion Modeling." CORROSION 75, no. 11 (September 7, 2019): 1285–99. http://dx.doi.org/10.5006/3282.

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The modeling of localized corrosion has usually focused on calculating the spatial and/or temporal distributions of chemical species, potential, and current. These are affected by the reactions considered, the geometry, and the modes of mass transport of importance. Finite element method (FEM) is a numerical technique to obtain approximate solutions to the differential equations based on different types of discretization in which the domain of interest is divided into different types of elements. The introduction of the FEM opened a variety of opportunities for increasing the complexity, and therefore the fidelity, of the localized corrosion conditions considered. This article first briefly introduces the FEM technique before describing the choices the modeler has with regards to the governing equations for the system. The history of the application of FEM to localized corrosion is given, highlighting the different aspects of localized corrosion that have been successfully modeled. Finally, some of the current challenges in FEM modeling of localized corrosion are outlined.

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Yoshida, Takumi, Takeshi Okuzono, and Kimihiro Sakagami. "Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration." Applied Sciences 10, no. 11 (May 28, 2020): 3750. http://dx.doi.org/10.3390/app10113750.

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This paper presents a proposal of a time domain room acoustic solver using novel fourth-order accurate explicit time domain finite element method (TD-FEM), with demonstration of its applicability for practical room acoustic problems. Although time domain wave acoustic methods have been extremely attractive in recent years as room acoustic design tools, a computationally efficient solver is demanded to reduce their overly large computational costs for practical applications. Earlier, the authors proposed an efficient room acoustic solver using explicit TD-FEM having fourth-order accuracy in both space and time using low-order discretization techniques. Nevertheless, this conventional method only achieves fourth-order accuracy in time when using only square or cubic elements. That achievement markedly impairs the benefits of FEM with geometrical flexibility. As described herein, that difficulty is solved by construction of a specially designed time-integration method for time discretization. The proposed method can use irregularly shaped elements while maintaining fourth-order accuracy in time without additional computational complexity compared to the conventional method. The dispersion and dissipation characteristics of the proposed method are examined respectively both theoretically and numerically. Moreover, the practicality of the method for solving room acoustic problems at kilohertz frequencies is presented via two numerical examples of acoustic simulations in a rectangular sound field including complex sound diffusers and in a complexly shaped concert hall.

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Kololikiye, Gilang Ramadan, Yulvi Zaika, and Harimurti Harimurti. "Prefabricatred Vertical Drain Improved Soft Soil Using Three-Dimensional Finite Element Method." Rekayasa Sipil 15, no. 2 (June 8, 2021): 150–56. http://dx.doi.org/10.21776/ub.rekayasasipil.2021.015.02.10.

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The subgrade layer of freeway construction in East Java contains high compressibility of soft soil with 24.5 m depth and 10.2 m height of the embankment. It is necessary to stabilize using PVD by accelerating the process of consolidation to increase its bearing capacity. In this study, 3D FEM programming is used to analyze the consolidation in pursuing to compare with the analytical results. 3D FEM shows the settlement without PVD is 0.834 m with excess pore water -4 kN/m2, while using PVD the settlement 0.819 m with excess pore pressure -8 kN/m2. For the analytical results, both variations indicate the settlement 0.787 m. Itâ€™s because the FEM discretization analyzed in 3D gives more accurate results.

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35

Elleithy, W. "FEM-BEM coupling for elasto-plastic analysis: automatic adaptive generation of the FEM and BEM zones of discretization." PAMM 7, no. 1 (December 2007): 2020053–54. http://dx.doi.org/10.1002/pamm.200700335.

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36

Hori, Muneo, Kenji Oguni, and Hide Sakaguchi. "Proposal of FEM implemented with particle discretization for analysis of failure phenomena." Journal of the Mechanics and Physics of Solids 53, no. 3 (March 2005): 681–703. http://dx.doi.org/10.1016/j.jmps.2004.08.005.

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37

Bui-Thanh, Tan, and Quoc P. Nguyen. "FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems." Inverse Problems and Imaging 10, no. 4 (October 2016): 943–75. http://dx.doi.org/10.3934/ipi.2016028.

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38

Bauer, Andrew C., and Abani K. Patra. "Performance of parallel preconditioners for adaptive hp FEM discretization of incompressible flows." Communications in Numerical Methods in Engineering 18, no. 5 (March 5, 2002): 305–13. http://dx.doi.org/10.1002/cnm.465.

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39

Liu, Guirong, Meng Chen, and Ming Li. "Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)." International Journal of Computational Methods 14, no. 04 (April 18, 2017): 1750036. http://dx.doi.org/10.1142/s0219876217500360.

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Анотація:

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

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40

Führer, Thomas, Norbert Heuer, Michael Karkulik, and Rodolfo Rodríguez. "Combining the DPG Method with Finite Elements." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 639–52. http://dx.doi.org/10.1515/cmam-2017-0041.

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AbstractWe propose and analyze a discretization scheme that combines the discontinuous Petrov–Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov–Galerkin) form with broken test space in one part, and of Bubnov–Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov–Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

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41

Sawicki, Dominik, and Eugeniusz Zieniuk. "Parametric Integral Equations Systems Method In Solving Unsteady Heat Transfer Problems For Laser Heated Materials." Acta Mechanica et Automatica 9, no. 3 (September 1, 2015): 167–72. http://dx.doi.org/10.1515/ama-2015-0028.

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Abstract One of the most popular applications of high power lasers is heating of the surface layer of a material, in order to change its properties. Numerical methods allow an easy and fast way to simulate the heating process inside of the material. The most popular numerical methods FEM and BEM, used to simulate this kind of processes have one fundamental defect, which is the necessity of discretization of the boundary or the domain. An alternative to avoid the mentioned problem are parametric integral equations systems (PIES), which do not require classical discretization of the boundary and the domain while being numerically solved. PIES method was previously used with success to solve steady-state problems, as well as transient heat transfer problems. The purpose of this paper is to test the efficacy of the PIES method with time discretization in solving problem of laser heating of a material, with different pulse shape approximation functions.

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42

Kielhorn, Lars, Thomas Rüberg, and Jürgen Zechner. "Simulation of electrical machines: a FEM-BEM coupling scheme." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 5 (September 4, 2017): 1540–51. http://dx.doi.org/10.1108/compel-02-2017-0061.

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Purpose Electrical machines commonly consist of moving and stationary parts. The field simulation of such devices can be demanding if the underlying numerical scheme is solely based on a domain discretization, such as in the case of the finite element method (FEM). This paper aims to present a coupling scheme based on FEM together with boundary element methods (BEMs) that neither hinges on re-meshing techniques nor deals with a special treatment of sliding interfaces. While the numerics are certainly more involved, the reward is obvious: the modeling costs decrease and the application engineer is provided with an easy-to-use, versatile and accurate simulation tool. Design/methodology/approach The authors present the implementation of a FEM-BEM coupling scheme in which the unbounded air region is handled by the BEM, while only the solid parts are discretized by the FEM. The BEM is a convenient tool to tackle unbounded exterior domains, as it is based on the discretization of boundary integral equations (BIEs) that are defined only on the surface of the computational domain. Hence, no meshing is required for the air region. Further, the BIEs fulfill the decay and radiation conditions of the electromagnetic fields such that no additional modeling errors occur. Findings This work presents an implementation of a FEM-BEM coupling scheme for electromagnetic field simulations. The coupling eliminates problems that are inherent to a pure FEM approach. In detail, the benefits of the FEM-BEM scheme are: the decay conditions are fulfilled exactly, no meshing of parts of the exterior air region is necessary and, most importantly, the handling of moving parts is incorporated in an intriguingly simple manner. The FEM-BEM formulation in conjunction with a state-of-the-art preconditioner demonstrates its potency. The numerical tests not only reveal an accurate convergence behavior but also prove the algorithm to be suitable for industrial applications. Originality/value The presented FEM-BEM scheme is a mathematically sound and robust implementation of a theoretical work presented a decade ago. For the application within an industrial context, the original work has been extended by higher-order schemes, periodic boundary conditions and an efficient treatment of moving parts. While not intended to be used under all circumstances, it represents a powerful tool in case that high accuracies together with simple mesh-handling facilities are required.

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43

Rastegari, Shafagh, Seyed Majid Hosseini, Mojtaba Hasani, and Abdolreza Jamilian. "An Overview of Basic Concepts of Finite Element Analysis and Its Applications in Orthodontics." Journal of Dentists 11 (July 5, 2023): 23–30. http://dx.doi.org/10.12974/2311-8695.2023.11.04.

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Purpose: The aim of this article is to acquaint the readers with the aims and goals of the finite element method and how to use it in dentistry and especially in orthodontics. Methods: The finite element method (FEM) has shown to be a beneficial research tool that has assisted scientists in various analyses such as stress-strain, heat transfer, dynamic, collision, and deformation analyses. The FEM is responsible for predicting the behavior of objects under different working conditions. It is a computational procedure to measure the stress in an element, which performs a model solution to solve a problem; the FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object. The technique of FEA lies in the development of a suitable mesh arrangement. Conclusions: The FEM can be effective in understanding the behavior of teeth, both jaws, craniofacial structure, and other hard tissue structures of humans under various working conditions, as the technique allows for evaluating tooth movement and the stress distribution within the surrounding alveolar bone, the periodontal ligament (PDL). This technique is exceptionally valuable for evaluating mechanical aspects of biomaterials and human tissues that can hardly be measured in vivo. This review article presents the FEM, its methodology, and its application in the orthodontic domain.

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Li, Mingxia, Jingzhi Li, and Shipeng Mao. "Numerical Analysis of an Adaptive FEM for Distributed Flux Reconstruction." Communications in Computational Physics 15, no. 4 (April 2014): 1068–90. http://dx.doi.org/10.4208/cicp.050313.210613s.

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AbstractThis paper studies convergence analysis of an adaptive finite element algorithm for numerical estimation of some unknown distributed flux in a stationary heat conduction system, namely recovering the unknown Neumann data on interior inaccessible boundary using Dirichlet measurement data on outer accessible boundary. Besides global upper and lower bounds established in [23], a posteriori local upper bounds and quasi-orthogonality results concerning the discretization errors of the state and adjoint variables are derived. Convergence and quasi-optimality of the proposed adaptive algorithm are rigorously proved. Numerical results are presented to illustrate the quasi-optimality of the proposed adaptive method.

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45

Löschner, Fabian, José Antonio Fernández-Fernández, Stefan Rhys Jeske, Andreas Longva, and Jan Bender. "Micropolar Elasticity in Physically-Based Animation." Proceedings of the ACM on Computer Graphics and Interactive Techniques 6, no. 3 (August 16, 2023): 1–24. http://dx.doi.org/10.1145/3606922.

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We explore micropolar materials for the simulation of volumetric deformable solids. In graphics, micropolar models have only been used in the form of one-dimensional Cosserat rods, where a rotating frame is attached to each material point on the one-dimensional centerline. By carrying this idea over to volumetric solids, every material point is associated with a microrotation, an independent degree of freedom that can be coupled to the displacement through a material's strain energy density. The additional degrees of freedom give us more control over bending and torsion modes of a material. We propose a new orthotropic micropolar curvature energy that allows us to make materials stiff to bending in specific directions. For the simulation of dynamic micropolar deformables we propose a novel incremental potential formulation with a consistent FEM discretization that is well suited for the use in physically-based animation. This allows us to easily couple micropolar deformables with dynamic collisions through a contact model inspired from the Incremental Potential Contact (IPC) approach. For the spatial discretization with FEM we discuss the challenges related to the rotational degrees of freedom and propose a scheme based on the interpolation of angular velocities followed by quaternion time integration at the quadrature points. In our evaluation we validate the consistency and accuracy of our discretization approach and demonstrate several compelling use cases for micropolar materials. This includes explicit control over bending and torsion stiffness, deformation through prescription of a volumetric curvature field and robust interaction of micropolar deformables with dynamic collisions.

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46

Skordaris, Georgios, Konstantinos Bouzakis, and Paschalis Charalampous. "A critical review of FEM models to simulate the nano-impact test on PVD coatings." MATEC Web of Conferences 188 (2018): 04017. http://dx.doi.org/10.1051/matecconf/201818804017.

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Nano-impact test is a reliable method for assessing the brittleness of PVD coatings with mono- or multi-layer structures. For the analytical description of this test, a 3D-FEM Finite Element Method (FEM) model and an axis-symmetrical one were developed using the ANSYS LS-DYNA software. The axis-symmetrical FEM simulation of the nano-impact test can lead to a significantly reduced computational time compared to a 3D-FEM model and increased result's accuracy due to the denser finite element discretization network. In order to create an axissymmetrical model, it was necessary to replace the cube corner indenter by an equivalent conical one with axis-symmetrical geometry. Results obtained by the developed FEM models simulating the nano-impact test on PVD coatings with various structures were compared with experimental ones. Taking into account the sufficient convergence between them as well as the reduced calculation time only in the case of an axis-symmetrical model, the latter introduced numerical procedure can be effectively employed to monitor the effect of various coating structures on their brittleness.

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47

Williams, F. W., and D. Kennedy. "Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames." International Journal of Structural Stability and Dynamics 03, no. 02 (June 2003): 299–305. http://dx.doi.org/10.1142/s0219455403000835.

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Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.

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Mavrič, Boštjan, and Božidar Šarler. "Application of the RBF collocation method to transient coupled thermoelasticity." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 5 (May 2, 2017): 1064–77. http://dx.doi.org/10.1108/hff-03-2016-0110.

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Purpose In this study, the authors aim to upgrade their previous developments of the local radial basis function collocation method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic-coupled thermoelasticity problems. Design/methodology/approach The authors solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multi-quadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton–Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference finite-element method (FEM) solution and, in one case, with a solution obtained using the meshless local Petrov–Galerkin method. Findings The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case. Research limitations/implications The performance of the LRBFCM near the tip of the shock front appears to be suboptimal because it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov–Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, the authors are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail. Originality/value For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity.

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Erath, Christoph, and Robert Schorr. "Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 251–72. http://dx.doi.org/10.1515/cmam-2018-0253.

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AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

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50

Цуканова, Екатерина, and Ekaterina Tsukanova. "Analysis of forced vibrations of frameworks by finite element method using dynamic finite element." Bulletin of Bryansk state technical university 2015, no. 2 (June 30, 2015): 93–103. http://dx.doi.org/10.12737/22911.

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The analysis of forced vibrations of frameworks using finite element method is considered. The dynamic finite element, the base functions of which represent exact dynamic shapes of structural elements, is used for system discretization. The assessment of errors as a result of classic FEM application is given. The efficiency of application of dynamic finite element for analysis of forced vibrations and dynamic stress-deformed state of structures is shown.

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