Global Algebraic System Research Articles

Modeling of Magnetoelectric Effects in Composite Structures by FEM–BEM Coupling

In this paper, a coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behavior of magnetoelectric effects in composite structures. This coupling of numerical methods makes it possible not to have to consider a free space domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns which is accompanied by shorter computation times compared to a classical FEM approach. A mixed magnetic vector potential, reduced magnetic scalar potential formulation is used for the magnetic problem, and classical FEM formulations are used for electrical and mechanical problems. The resulting global algebraic system is solved by a block Gauss-Seidel solver.

A Prediction–Correction Solver for Real-Time Simulation of Free-Surface Flows in River Networks

A prediction–correction solver is presented here for rapid simulation of free-surface flows in dendritic and looped river networks. Rather than solving a large global algebraic system over the entire domain of river networks, the model solves subsystems for subdomains of branches in two steps: prediction and correction. With the help of the prediction–correction method (PCM), the partial linearization technique, the semi-implicit method, and the Eulerian–Lagrangian method (ELM), the model only needs to solve tridiagonal linear systems for branches and is free of any iteration. The new model was tested using a hypothetical looped river network with regular cross sections, the Three Gorges Reservoir (TGR) dendritic river network, and the Jing-south looped river system (with seasonally flooding branches). In the first test, a time-step sensitivity study was conducted and the model was revealed to produce accurate simulations at large time steps when the condition for application of the PCM to river networks was satisfied. In the TGR test, the PCM model provided almost the same histories of water levels and discharges as those simulated by the HEC-RAS model. In the Jing-south test, the mean absolute error in simulated water levels was 0.07–0.24 m, and the relative error in simulated cross-section water flux was 0.5–4.9% compared with field data; the conservation error was generally 2 × 10−4 to 3 × 10−4. The PCM model was revealed to be 2–4 times as fast as a reported model, which solves local nonlinear subsystems using two-layer iterations, and 1.2–1.4 times as fast as the HEC-RAS. Using a time step of 1200 s, it took the sequential code 26.8 and 23.1 s to complete a simulation of a one-year unsteady flow process, respectively, in the TGR river networks (with 588 cells) and the Jing-south river system (with 662 cells).

Element differential method for free and forced vibration analysis for solids

Abstracts In this work a new strong-form numerical method, element differential method (EDM), is proposed to perform free and forced vibration analysis of elastodynamic problems. The present method establishes the global algebraic system equations directly based on the strong form of the equilibrium equations without using any variational principles or energy principles. In this method, the isoparametric elements with a node inside them are utilized to discrete the geometries. And the direct differentiation of their shape functions is used to characterize the geometry and physical variables. A novel collocation technique is then proposed to generate the system of equations, in which the dynamic equilibrium equations are collocated only at the internal nodes of elements, and the traction equilibrium equations are collocated at the interface and outer surface nodes. Compared with the standard finite element method, no integrals are involved to form the coefficients of the system and a reduced and lamped mass matrix can be directly obtained from the density properties of the target problem. Since the mass term only exists at the internal nodes of elements, the dynamic coupling of final system equations of the structure can be reduced, which will greatly save the computational resources. Numerical examples about free and forced 2D and 3D dynamic problems are given to demonstrate the correctness and efficiency of the proposed method.

Solution of skin-effect problems by means of the hybrid SDBCI method

Purpose – The purpose of this paper is to present a modified version of the hybrid Finite Element Method-Dirichlet Boundary Condition Iteration method for the solution of open-boundary skin effect problems. Design/methodology/approach – The modification consists of overlapping the truncation and the integration boundaries of the standard method, so that the integral equation becomes singular as in the well-known Finite Element Method-Boundary Element Method (FEM-BEM) method. The new method is called FEM-SDBCI. Assuming an unknown Dirichlet condition on the truncation boundary, the global algebraic system is constituted by the sparse FEM equations and by the dense integral equations, in which singularities arise. Analytical formulas are provided to compute these singular integrals. The global system is solved by means of a Generalized Minimal Residual iterative procedure. Findings – The proposed method leads to slightly less accurate numerical results than FEM-BEM, but the latter requires much more computing time. Practical implications – Then FEM-SDBCI appears more appropriate than FEM-BEM for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Originality/value – Note that FEM-SDBCI assumes a Dirichlet condition on the truncation boundary, whereas FEM-BEM assumes a Neumann one.

Strong formulation finite element method for arbitrarily shaped laminated plates - Part I. Theoretical analysis

This paper provides a new technique for solving the static analysis of arbitrarily shaped composite plates by using Strong Formulation Finite Element Method (SFEM). Several papers in literature by the authors have presented the proposed technique as an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The present methodology joins the high accuracy of the strong formulation with the versatility of the well-known Finite Element Method (FEM). The continuity conditions among the elements is carried out by the compatibility or continuity conditions. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is easy and straightforward. The main novelty of this paper is the application of the stress and strain recovery once the displacement parameters are evaluated. Computer investigations concerning a large number of composite plates have been carried out. SFEM results are compared with those presented in literature and a perfect agreement is observed.

Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes

This paper provides a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM). The proposed technique, also known as Multi-Domain Differential Quadrature (MDQ), is an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The multi-domain method can be directly applied to regular sub-domains of rectangular shape, as well as to elements of general shape when a coordinate transformation is considered. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the parent space, called computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is simple and straightforward. Computer investigations concerning a large number of membrane geometries have been carried out. GDQFEM results are compared with those presented in the literature and a perfect agreement is observed. Membranes of complex geometry with a material inhomogeneity are also carefully examined. Numerical results referring to some new unpublished geometric shapes are reported to let comparisons with further research on this subject.

GMRES Solution of FEM-BEM Global Systems for Electrostatic Problems Without Voltaged Conductors

This paper extends an iterative method to efficiently solve the global algebraic system of equations obtained by applying the hybrid FEM-BEM method to cases in which the field is due to charge distributions only, that is, when conductors with assigned potential (Dirichlet conditions) are not present. In the proposed approach, a conjugate gradient solver is used to solve the singular FEM equations, whereas the BEM equations are modified by means of the Gauss law.

A Modified FEM-DBCI Method for Static and Quasi-Static Electromagnetic Field Problems

This paper presents a modified version of the hybrid FEM-DBCI (Finite Element Method-Dirichlet Boundary Condition Iteration) method to solve static and quasi-static electromagnetic field problems in open boundary domains. The modification consists of overlapping the fictitious truncation boundary with the integration one, as in the FEM-BEM (Boundary Element Method) method. The global algebraic system obtained is solved iteratively by means of the GMRES (Generalized Minimal Residual) solver, applied virtually to a reduced system in which the unknowns are those relative to the potential on the truncation boundary only. Some validation examples are provided.

Thermal analysis of an eddy‐current heated piece by means of the FEM‐DBCI method

PurposeThis paper aims to propose a hybrid method, called finite element method‐Dirichlet boundary condition iteration (FEM‐DBCI), for the computation of time‐harmonic eddy current problems inside a conductor heated by coils in 3D open‐boundary geometry.Design/methodology/approachThe method assumes the electrical field as unknown on a mesh of tetrahedral edge elements. The heating power density inside the conductor is then computed and a steady‐state thermal analysis is performed on the same mesh of nodal tetrahedra to calculate the temperature distribution inside the heated piece, taking radiation and convection into account. A numerical example is also provided.FindingsThe method couples a differential equation for the interior problem in terms of the electric fields with an integral equation for the exterior one. The global algebraic system is efficiently solved in an iterative way.Originality/valueThe paper illustrates the computation of time‐harmonic eddy current problems inside a conductor heated by coils.

An Iterative Solution to FEM-BEM Algebraic Systems for Open-Boundary Electrostatic Problems

This paper describes an iterative method to solve efficiently the global algebraic system of equations obtained by applying the hybrid finite element method-boundary element method (FEM-BEM) to the solution of open-boundary electrostatic problems. In the proposed approach, a conjugate gradient (CG) solver is used to solve the FEM equations, whereas the BEM equations are solved by making use of the direct LU solver. To do so, it is convenient to implement the BEM equations in a nonconventional way, by choosing the nodes of the potential as not coinciding with those of its normal derivative

Finite-element solution of eddy-current problems in unbounded domains by means of the hybrid FEM-DBCI method

A hybrid method, called finite-element method-Dirichlet boundary-condition iteration (FEM-DBCI), is described for the finite-element solution of time-harmonic eddy current problems in three-dimensional unbounded domains. The procedure couples a differential equation for the interior problem in terms of the electric field with an integral equation for the exterior one, which expresses the Dirichlet condition on the truncation boundary. The global algebraic system is efficiently solved in an iterative way. FEM-DBCI leads to shorter solving times and not much higher memory requirements with respect to the FEM-boundary-element method.

Quadratic, streamline upwinding for finite element method solutions to 2-D convective transport problems

During the past decade, Finite Element Methods (FEMs) have been recognized to be more powerful tools in the solution of various flow problems as opposed to their predecessors, Finite Difference Methods. On a fundamental level, the FEM involves approximations to the solution over a given trial space on a discrete mesh, rather than Finite Difference approximations to the differential operators on a discrete mesh. In this way, the mechanics (operators) of the FEM remain closer to the physical principles associated with a given problem. In addition to this, the FEM incorporates boundary conditions very efficiently into the numerical formulation. Much current research is aimed at making these FEMs viable options for convection-dominated flows. In 1985, Mizukami et al. published two papers advancing the approach of streamline upwinding within linear elements for FEMs. The purpose of this work is to demonstrate that streamline upwinding of higher ordered elements is a more accurate option. The work goes on to show exactly how streamline upwinding effects the stability of iteratively solving the global algebraic system resulting from the FEM.

Mutual finite-element universal matrices in skin effect problems

Mutual finite-element universal matrices are defined for the numerical solution of a certain class of integrodifferential equations. The term 'mutual' comes from the fact that these matrices are relative to a couple of finite elements. By using these matrices an extension of the classical per-element assembly procedure of the global algebraic system is easily implemented. An example of application to the skin effect analysis of shielded multiconductor transmission lines is given. The proposed approach seems very promising because it allows easy inclusion of integrodifferential formulation into general-purpose N-dimensional finite-element codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>