Nodal Finite Element Research Articles

A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian–Eulerian computations with nodal finite elements

This article describes a conservative synchronized remap algorithm applicable to arbitrary Lagrangian–Eulerian computations with nodal finite elements. In the proposed approach, ideas derived from flux-corrected transport (FCT) methods are extended to conservative remap. Unique to the proposed method is the direct incorporation of the geometric conservation law (GCL) in the resulting numerical scheme. It is shown here that the geometric conservation law allows the method to inherit the positivity preserving and local extrema diminishing (LED) properties typical of FCT schemes. The proposed framework is extended to the systems of equations that typically arise in meteorological and compressible flow computations. The proposed algorithm remaps the vector fields associated with these problems by means of a synchronized strategy. The present paper also complements and extends the work of the second author on nodal-based methods for shock hydrodynamics, delivering a fully integrated suite of Lagrangian/remap algorithms for computations of compressible materials under extreme load conditions. Extensive testing in one, two, and three dimensions shows that the method is robust and accurate under typical computational scenarios.

Accelerating the Convergence of Algebraic Multigrid for Quadratic Finite Element Method by Using Grid Information and p-Multigrid

This paper proposes a new multigrid method to efficiently solve the finite element approximation of static or quasi-static electromagnetic problems with quadratic nodal finite elements on unstructured grids in 2D and 3D by using grid information and p-multigrid. Unlike the traditional algebraic multigrid (AMG), the new method constructs the first level coarse grid from the grid information of the finest grid directly, by using the natural geometrical coarsening relationship between the P2 and P1 elements. Then, the algebraic equations of the first level coarse grid are constructed based on the special relationship between the basis functions of quadratic finite element and its linear counterpart. At last, the traditional AMG is applied to solve the algebraic equations of the first coarse level rather than that of the finest grid. Several techniques of convenient and economic implementation are discussed. For the problems tested, the proposed method is much more efficient than the conjugate gradient method with incomplete Cholesky preconditioning; in addition, compared with traditional Krylov subspace accelerated AMG, the new method may save about 30% to 40% CPU time while achieving the same accuracy in practical computations.

Efficient Arnoldi-type algorithms for rational eigenvalue problems arising in fluid–solid systems

We develop and analyze efficient methods for computing damped vibration modes of an acoustic fluid confined in a cavity with absorbing walls capable of dissipating acoustic energy. The discretization in terms of pressure nodal finite elements gives rise to a rational eigenvalue problem. Numerical evidence shows that there are no spurious eigenmodes for such discretization and also confirms that the discretization based on nodal pressures is much more efficient than that based on Raviart–Thomas finite elements for the displacement field. The trimmed linearization method is used to linearize the associated rational eigenvalue problem into a generalized eigenvalue problem (GEP) of the form A x = λ B x . For solving the GEP we apply Arnoldi algorithm to two different types of single matrices B - 1 A and AB - 1 . Numerical accuracy shows that the application of Arnoldi on AB - 1 is better than that on B - 1 A .

A nodal position finite element method for plane elastic problems

This paper presents a new nodal position finite element method (NPFEM) as an alternative to the existing finite element method (FEM) for plane elastic problem. The newly developed method addresses the complications of the existing FEM in dealing with dynamic problems experiencing large rigid-body motion coupled with small elastic deformation. Unlike the existing FEM that is based on nodal displacements, the new NPFEM uses nodal positions as basic variables to eliminate the need to decouple the elastic deformation from the rigid-body motion. As a result, it can avoid the accumulated errors arising from the existing FEM over a long period of time by comparing the deformed element with its undeformed status directly. This will be very useful in dynamic modeling of mechanical system where the current positions of parts are more meaningful to designers than the displacements. In addition, the new NPFEM has the potential to address the need in bridging the position based molecular dynamics (MD) to the displacement based finite element (FE) modeling in the multiscale MD/FE analysis. Thus, the NPFEM can provide a unified description in multiscale MD/FE modeling in future.

Nodal finite element approximations for the neutron transport equation

Two classes of nodal methods, weakly and strongly discontinuous, are introduced and applied to the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry and discrete ordinates. These methods are then applied for the approximation of the solution of a reference problem well known in the nuclear engineering literature.

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.

An 𝐸-based mixed formulation for a time-dependent eddy current problem

In this paper, we analyze a mixed form of a time-dependent eddy current problem formulated in terms of the electric fieldE\boldsymbol {E}. We show that this formulation admits a well-posed saddle point structure when the constraints satisfied by the primary unknown in the dielectric material are handled by means of a Lagrange multiplier. We use Nédélec edge elements and standard nodal finite elements to define a semi-discrete Galerkin scheme. Furthermore, we introduce the corresponding backward-Euler fully-discrete formulation and prove error estimates.

Comparison of Finite-Difference and Finite-Element Schemes for Magnetization Processes in 3-D Particles

This paper compares two numerical schemes for the simulation of magnetization dynamics in 3-D particles. The first one is based on the finite-difference computation of the Landau-Lifshitz equation and on the magnetostatic field evaluation by fast Fourier transforms. The second one handles the spatial discretization of the effective field expression by a nodal finite-element method and solves the Poisson equation with a finite-element/boundary-element technique. The convergence of the methods is studied by varying the time and space discretization.

A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems

In this paper we propose a method for solving the electro-magnetostatics problem and the eddy current problem in terms of suitable potentials. A new variational formulation is devised, in which standard results of potential theory are used to reduce the problem in the external domain to an integral equation on the boundary of a computational domain containing the conductor. The existence and uniqueness of the solution is proved, by showing that the associated sesquilinear form is coercive. A numerical approximation scheme, based on nodal finite elements in the computational domain and boundary elements on its boundary, is devised and proved to be convergent. It is also shown that the solution of the time-harmonic eddy current problem tends to the solution of the electro-magnetostatics problem as the frequency tends to 0. The same convergence holds, uniformly with respect to the mesh size, for the finite element solutions.

A Finite Element Procedure for Dynamic Micromagnetic Computations

This paper presents a 3-D dynamic micromagnetic numerical scheme, based on the decoupling of the effective field evaluation from the treatment of the Landau-Lifshitz equation. The weak formulation of the effective field expression is spatially discretized through a nodal finite element approach. The micromagnetic equation is locally computed, performing the time integration with a finite difference scheme, based on midpoint rule and able to preserve the magnetization nonconvex constraint. Different techniques are compared for the evaluation of the magnetostatic field, such as finite element/boundary element method, standard finite element approach with first-order asymptotic boundary conditions, and Green integral formulation.

Nodal and vector finite elements in static and eddy current field problems

The numerical analysis or design of an arrangement, which require electromagnetic field calculation, can be characterized by the electric and magnetic field intensities and flux densities. For determination of these field quantities in the electromagnetic field, one method is to find the solution of the partial differential equations of the field quantities under prescribed boundary conditions obtained from Maxwell’s equations. The Finite Element Method (FEM) is a possible technique to solve partial differential equations, which is based on the weak form of the weighted residual method. The paper presents some potential formulations, which can be used for solving static magnetic field problems and eddy current field problems with the help of FEM. Some examples are also presented at the second part of the paper.

Reuse, Recycle, Reduce (3R) – strategies for the calculation of transient magnetic fields

The discretization of transient magneto-dynamic field problems with geometric discretization schemes such as the Finite Integration Technique or the Finite-Element Method based on Whitney form functions results in nonlinear differential-algebraic systems of equations of index 1. Their time integration with embedded s-stage singly diagonal implicit Runge–Kutta methods requires the solution of s nonlinear systems within one time step. Accelerated solution of these schemes is achieved with techniques following so-called 3R-strategies (“ reuse, recycle, reduce”). This involves e.g. the solution of the linear(-ized) equations in each time step where the solution process of the iterative preconditioned conjugate gradient method reuses and recycles spectral information of linear systems from previous stages. Additionally, a combination of an error controlled spatial adaptivity and an error controlled implicit Runge–Kutta scheme is used to reduce the number of unknowns for the algebraic problems effectively and to avoid unnecessary fine grid resolutions both in space and time. First numerical results for 2D nonlinear magneto-dynamic problems validate the presented approach and its implementation. The space discretization in the numerical examples is done by Lagrangian nodal finite elements but the presented algorithms also work in combination with other discretization schemes for the Maxwell equations such as the Whitney vector finite elements.

Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method

AbstractNon‐overlapping domain decomposition techniques are used to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the so‐called cross‐points, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter defining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the cross‐points and illustrating the strategy to accelerate the iterative procedure are presented. Copyright © 2007 John Wiley & Sons, Ltd.

Parallel implementation of a three-dimensional electrostatic PIC–MCC method using an unstructured tetrahedral mesh

A parallel three-dimensional electrostatic particle-in-cell Monte Carlo Collision (PIC-MCC) code using an unstructured tetrahedral mesh is developed and validated in this paper. Poisson's equation is discretized by a nodal finite element method using a linear interpolation function. Charmed-particle motion is traced cell by cell using a leap-frog method and the Boris scheme. Code is validated using quasi-one-dimensional radiofrequeney gas discharge by observing the bi-Maxwellian elecfrom energy probability distribution function at low pressure. Parallel performance of the PIC code is tested using a three-dimensional radiofrequency gas discharge on a PC-cluster system. Results show that parallel efficiency of 83% can be achieved at 32 processors with dynamic domain decomposition.

Biorthogonal bases with local support and approximation properties

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p - 1. Working with Gaus-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.

A-θ approaches of an eddy current problem based on solving certain potentials

In this paper, we propose novel finite element A - θ approaches to realize the approximation of a three-dimensional eddy current problem in a bounded domain containing a simply-connected conductor region where the eddy current is formed. These algorithms are based on solving a vector potential in the whole domain by using the edge finite-element method and a local scalar potential only in the conductor region by using the nodal finite-element method. Further, error estimates for coupled and decoupled A - θ schemes are given.

Mesh-centered finite differences from unconventional mixed-hybrid nodal finite elements

It is shown how mesh-centered finite differences can be obtained from unconventional mixed-hybrid nodal finite elements. The classical Raviart-Thomas schemes of index k (RTk) are based on interpolation parameters that are cell and/or edge moments. For the unconventional form (URTk), they become point values at Gaussian points. In particular, the scheme URT1 is fully described. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006

An E-based splitting finite element method for time-dependent eddy current equations

A new decoupled finite element method is suggested to approximate time-dependent eddy current equations in a three-dimensional polyhedral domain. This method is based on solving a vector and a scalar from the splitting of the electric field by using edge and nodal finite elements. An optimal energy-norm error estimate in finite time is obtained by introducing a projection operator.

Simulation of the quasi‐static electric field by an electric current vector potential approximated by edge elements

PurposeThe aim of the present work is to find an efficient solution concerning the computational effort of quasi‐static electric field (QSEF) problems involving anisotropic conductivity and permittivity in the frequency domain.Design/methodology/approachNumerical simulations are carried out with tetrahedral nodal finite elements of first‐ and second‐order and with Withney elements. The solution of the boundary value problem with the aid of the electric scalar potential approximated by nodal finite elements is compared with those by the electric current vector potential represented by edge finite elements.FindingsThe simulation with an electric current vector potential approximated by the edge elements of first‐order prevail over that by the electric scalar potential approximated by nodal elements of second‐order concerning the memory requirements and the computation time at comparable accuracy.Originality/valueThe application of edge finite elements to solve QSEF problems considering an anisotropic complex conductivity in 3D.