Vector Shape Functions Research Articles

3D MCSEM modeling based on domain-decomposition finite-element method

Domain decomposition approach is effective in converting large modeling problems into many small ones, so that the parallel computation can be easily facilitated and the memory consumption can be largely reduced. In this paper, we developed a domain-decomposition method for 3D time-domain marine controlled-source (MCSEM) modeling based on Dual-Primal Finite-Element Tearing and Interconnecting technique (FETI-DP). We use tetrahedral grids to do the spatial discretization first and then partition the mesh into subdomains. In each subdomain, the finite-element equations are independently constructed using the first-order vector shape functions and boundary conditions. To build the interconnections among these subdomains, we establish the interface equation for the Lagrange multipliers that are used as the boundary conditions for the subdomains. After solving the equations, the electric field for all subdomains can be obtained from the Lagrange multipliers. For the time discretization, we use unconditionally stable backward Euler scheme. We validate our algorithm by comparing with a semi-analytical solutions for a uniform half-space model under the ocean. Numerical experiments show that compared to the conventional finite-element method, our domain-decomposed method can save large amount of memory, render it possible to run 3D MCSEM modeling in small computational equipment.

An efficient construction of divergence-free spaces in the context of exact finite element de Rham sequences

Exact finite element de Rham subcomplexes relate conforming subspaces in H1(Ω), H(curl;Ω), H(div;Ω), and L2(Ω) in a simple way by means of differential operators (gradient, curl, and divergence). The characteristics of such strong couplings are crucial for the design of stable and conservative discretizations of mixed formulations for a variety of multiphysics systems. This work explores these aspects for the construction of divergence-free vector shape functions in a robust fashion allowing stable and faster simulations of mixed formulations of incompressible porous media flows. The resulting schemes are verified by means of numerical tests with known smooth solutions and applied to a benchmark problem to confirm the expected theoretical and computational performance results.

Creating higher order vector shape functions based on H(curl) for the edge meshless method

ABSTRACT The goal of this work is to extend the Edge Meshless Method (EMM) presenting a new way to build the vector shape functions based on the H(curl) space. These vector shape functions allow the use of four, five and six edges in the support domain. For this, the basis functions polynomial order is increased. The new EMM vector shape functions are applied to eigenvalues problems with different media. The numerical solution is not corrupted by spurious modes, field singularities generated by corners are correctly overcome and the continuity of tangential components across the interface between two different media is satisfied. The shape function with six edges in the support domain present the best results, where the majority of the eigenvalues double their convergence rate. Also, approximations of different polynomial orders can be used in the same problem.

Nodal Meshless Method With Vectorial Shape Functions Based on H(Curl)

The aim of this article is to deepen the idea of vectorial meshless methods, creating a nodal meshless method for vector problems, which we will call the vector nodal meshless method. In this proposal, a set of nodes is distributed in the domain and on its boundary, and each node has an associated unit vector. The vector shape functions are based on the H(curl) spaces and Nedelec’s first type elements polynomial space. The method is applied in the solution of the eigenvalues problem on waveguides filled with different materials. The numerical solution is not corrupted by spurious modes, it satisfies the tangential field continuity at the interface among different materials, and it overcomes problems of field singularities caused by corners and edges of the geometry.

Frequency-based investigation of a floating wave energy converter system with multiple flaps

In the present paper, a new Floating Wave Energy Converter System (FloWECS), consisting of a floating platform and multiple rotating flaps, is proposed and is preliminary investigated. The numerical analysis is implemented in the frequency domain under the action of regular head and oblique incident waves and it is based on a “dry” mode superposition approach. In this approach, the generalized modes concept is utilized for describing the relative to the platform rotation of the flaps, additionally to the six rigid-body modes. The required mode shapes of the generalized modes are determined through appropriate vector shape functions, which are derived in the present paper. The diffraction/radiation problem is solved using a numerical model based on the conventional boundary integral equation method. The proposed numerical formulation is, initially, validated by comparing results with the numerical ones of other investigators for the case of a single, bottom-hinged, rotating plate. Next, the exciting forces and the response of the “isolated” platform are assessed, while, finally, the hydrodynamic behaviour and the energy absorption of the FloWECS are investigated. The results illustrate, that for both examined incident wave directions, the generalized modes’ response and, thus, the system's power absorption is characterized by the existence of distinctive peaks at two different wave frequency ranges, attributed to resonance effects, as well as to the strong interaction effects between the activated flaps.

A Novel Vector Hysteresis Model Using Anisotropic Vector Play Model Taking Into Account Rotating Magnetic Fields

This paper presents an improved identification algorithm of the scalar play model, and suggests a novel vector play model to describe the vector hysteretic behavior of a nonoriented electrical steel sheet (ESS) under rotating magnetic fields. The proposed identification algorithm gives much better numerical accuracy than a conventional one by assigning more play operators than the measured asymmetric minor loops. The vector-shape function of the suggested vector play model consists of two scalar shape functions, which are independently identified by using measured hysteresis loops under alternating magnetic fields along rolling and transverse direction of ESS, respectively. The vector play model, furthermore, employs a nonlinear coefficient, which is a function of the magnitude and direction of magnetic flux density, to increase the numerical accuracy. The validity of the suggested vector play model is verified by comparing it with experimental results under various magnetic-field conditions.

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

SummaryA new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.

Vector Interpolation on Natural Element Method

An electromagnetic simulation often requires vector edge-based shape functions. In a finite-element method (FEM) context, vector shape functions are designed for keeping the tangential continuities and the normal discontinuities between elements, which correspond to the physical nature of electromagnetic fields. However, in the meshless methods context, the construction of vector shape functions is not direct and a very little work has been published on a vector interpolation. Presenting the typical characteristics of meshless methods, the natural element method (NEM) has proved to be able to provide more accurate and computationally efficient solutions than the standard FEM for node-based approximations. This paper is the first proposal of an NEM-based vector interpolation for polygonal/polyhedral cells. Results show that the implemented approach provides better accuracy than the edge-based FEM for a given number of degrees of freedom.

Anisotropic Vector Play Model Incorporating Decomposed Shape Functions

We propose an accurate anisotropic vector play model that uses the decomposition of vector shape functions. The parallel and perpendicular components of the magnetic field ${H}$ to the magnetic flux density ${B}$ are independently identified in constructing the decomposed shape functions. We further propose a method to reconstruct the perpendicular component from the 1-D measurement of the parallel component based on a magnetic energy approach. This method accurately reconstructs the vector hysteretic properties of silicon steel under alternating and rotating magnetic flux conditions.

Overview of the Finite Element Method

By using scalar and vector potentials, Maxwell's equations can be transformed into partial differential equations. Generally, the partial differential equations can be solved by numerical methods. One of these numerical methods is the finite element method, which is based on the weak formulation of the partial differential equations. The basis of numerical techniques is to reduce the partial differential equations to algebraic ones whose solution gives an approximation of the unknown potentials and electromagnetic field quantities. This reduction can be done by discretizing the partial differential equations in time if necessary and in space. The potential functions, the approximation method and the generated mesh distinguish the numerical field solvers. This paper summarize the finite element method as a CAD technique in electrical engineering to obtain the electromagnetic field quantities in the case of static magnetic field and eddy current field problems. Here, we show how to discretize the analyzed domain with finite elements, how to approximate potential functions with nodal and vector shape functions and how to build up the system of equations, which solution obtain the unknown potentials.

An edge element approach for dynamic micromagnetic modeling

This paper proposes a three-dimensional dynamic micromagnetic model, based on the Galerkin weak formulation, reconstructing magnetization by finite element edge vector shape functions. The demagnetizing filed is computed using a hybrid finite element boundary element method. The procedure is compared to analytical formulas and simulations performed with the NIST/OOMMF code, focusing on damping and precessional switching in magnetic thin films.

Using the Loki 3D edge-finite-element program to model EM dipole-dipole drill-hole data

Electromagnetic drill-hole measurements are an important tool for locating conductive orebodies in detailed exploration scale. Dipole-dipole systems have good spatial resolution, especially if all three magnetic components are measured over a wide frequency range. Interpretation of field data requires the use of sound 3D modelling tools capable of dealing with high conductivity contrasts and the close proximity of boundaries to the source or receiver. We modelled the response of the SlimBoris dipole-dipole drill hole system going through a block target using Loki, a 3D, full-domain, edge finite-element approach based on vector shape functions and scalar unknowns. This method achieves considerable speed advantage over conventional finite-element methods since only the one tangential component rather than three orthogonal components need be solved at each ``node'. Moreover, since no boundary conditions are violated in this approach, contrasts in excess of one million to one can be modelled accurately as verified through semi-analytical layered earth solutions. Computation time can be reduced by a further factor of five by solving initially for Schelkunoff potentials rather than electric or magnetic fields. This is due to the superior condition number of the resulting matrices. Accuracy is maintained using Green?s function projectors to obtain the fields at the receivers rather than differentiating the potentials. A 30,000 cell model takes 20 seconds on a 1.9GHz Intel chip per frequency per transmitter position. Control files for complex models can be set up rapidly using either the Encom EMGUI or Maxwell from EMIT. The results were in close agreement with scale model results at all contrast ranges. Another check was made using a 3D integral equation program. As expected, close agreement was obtained at contrasts of less than 300 but the integral equation results deteriorated at high contrast.

212 GSMAC 有限要素法による電磁熱流体解析 : ベクトル形状関数の誘導方程式への適用

212 GSMAC 有限要素法による電磁熱流体解析 : ベクトル形状関数の誘導方程式への適用

Shape Functions for Velocity Interpolation in General Hexahedral Cells

Numerical methods for grids with irregular cells require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element (CVMFE) methods, vector shape functions approximate velocities and vector test functions enforce a discrete form of Darcy's law. In this paper, a new vector shape function is developed for use with irregular, hexahedral cells (trilinear images of cubes). It interpolates velocities and fluxes quadratically, because as shown here, the usual Piola-transformed shape functions, which interpolate linearly, cannot match uniform flow on general hexahedral cells. Truncation-error estimates for the shape function are demonstrated. CVMFE simulations of uniform and non-uniform flow with irregular meshes show first- and second-order convergence of fluxes in the L2 norm in the presence and absence of singularities, respectively.

Systematic construction of three-dimensional ultra high-order Nedelec's elements

A new systematic construction of the three-dimensional ultra high-order Nedelec's elements is presented in this paper. It is proposed that a set of vector basis functions be simply chosen out of irrotational and rotational vector polynomials. Vector shape functions of the element are uniquely derived by the conditions modified from the scalar elements. The significance of the positions and directions of the arrows in the Nedelec's element is described. The higher order elements are also implemented in a cavity analysis.

Multiparametric vector finite elements: a systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions

A systematic methodology for the construction of three dimensional tetrahedral higher order vector finite elements is presented. Appropriate degrees of freedom describing tangential field components on interfaces and shape functions depending on a set of unknown coefficients are introduced. A decoupling procedure is applied to determine the shape functions. This leads to a whole family of parameter dependent vector finite elements. However, concerning the irrotational fields to be modeled by the finite element, additional relations among the coefficients are introduced, which fully determines them. The whole procedure is described via the introduction of a new second order tetrahedral finite element. Third order tangential vector finite elements are also derived and the method could be easily generalized to higher orders. A numerical implementation of the second order element in a waveguide scattering problem is finally presented.

A SYSTEMATIC APPROACH TO THE CONSTRUCTION OF HIGHER ORDER VECTOR FINITE ELEMENTS FOR THREE‐DIMENSIONAL ELECTROMAGNETIC FIELD COMPUTATION

The importance of vector finite elements for electromagnetic field computation has been extensively demonstrated in the recent literature. In this study we present a systematic approach to the construction of higher order vector shape functions based on dual vector expansions. The methodology presented results in explicit forms for second order elements. Moreover, it reveals the presence of a whole family of vector finite elements depending on the choice of the degrees of freedom and a set of arbitrary parameters.

Utilization of geometric symmetries in 'edge' based boundary integral eddy current solutions

An efficient implementation of this simple layer integral equation procedure for eddy current computation is presented. Vector boundary elements of quadrilateral and trilateral shapes are constructed. These elements, based on facet finite elements, have unknowns associated with edges, and are useful for modeling vector surface currents. The simple layer integral procedure is then discretized via the method of moments. A procedure is implemented to fully utilize geometric symmetries of the interior domain. The symmetry planes are not necessarily perfect electric or magnetic conductors. Using symmetric vector shape functions and introducing symmetric components allows the coefficient matrix to be symmetrized and then reduced to smaller systems. This procedure is used to compute the eddy current profile of a conducting plate with a small slot. >

A new displacement bound technique for rigid plastic continua subjectet to strong dynamic loading at large deflections II.

The method developed in [ 1] is applied herein in order to obtain lower hound to large displacements of the hinge supported circular plate acted on by impulsively, uniformly distributed loading. The plate is made from rigid and perfectly plastic material which yields according to the yield condition shown in fig- 1· The lower bound (1.21) is obtained when the dynamically admissible displacement and velocity are chosen in separated variable form which is a scalar time function multiplied by a vector shape function of space variable (1.13). In the case when W, Ẇ * are chosen in form (1.22). [comparison of the obtained estimate Eq (1.24) with previous solution of the same problem [1], upper bound [2] and experimental date [3] are presented in fig. 2