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The space discretization of eddy‐current problems in the magnetic vector potential formulation leads to a system of differential‐algebraic equations. They are typically time discretized by an implicit method. This requires the solution of large linear systems in the Newton iterations. The authors seek to speed up this procedure. In most relevant applications, several materials are non‐conducting and behave linearly, e.g. air and insulation materials. The corresponding matrix system parts remain constant but are repeatedly solved during Newton iterations and time‐stepping routines. The paper aims to exploit invariant matrix parts to accelerate the system solution.

Following the principle “reduce, reuse, recycle”, the paper proposes a Schur complement method to precompute a factorization of the linear parts. In 3D models this decomposition requires a regularization in non‐conductive regions. Therefore, the grad‐div regularization is revisited and tailored such that it takes anisotropies into account.

The reduced problem exhibits a decreased effective condition number. Thus, fewer preconditioned conjugate gradient iterations are necessary. Numerical examples show a decrease of the overall simulation time, if the step size is small enough. 3D simulations with large time step sizes might not benefit from this approach, because the better condition does not compensate for the computational costs of the direct solvers used for the Schur complement. The combination of the Schur approach with other more sophisticated preconditioners or multigrid solvers is subject to current research.

The Schur complement method is adapted for the eddy‐current problem. Therefore, a new partitioning approach into linear/non‐linear and static/dynamic domains is proposed. Furthermore, a new variant of the grad‐div gauging is introduced that allows for anisotropies and enables the Schur complement method in 3D.

The purpose of this paper is to develop a coupled approach to solve the pressure–velocity-coupled problem efficiently in the three-dimensional injection molding simulation.

A fully coupled pressure–velocity algorithm is developed to solve the coupled problem, by treating the pressure gradient term implicitly. And, the Schur complement preconditioned FGMRES is applied to decompose the resulting coupled pressure–velocity equation into pressure and velocity subsystems. Then, BoomerAMG is adopted to solve the pressure subsystem, and block Jacobi preconditioned FGMRES is applied to the velocity subsystem.

According to the several experiments, the fully coupled pressure–velocity algorithm was demonstrated to have faster convergence than the traditional SIMPLE algorithm, and the calculating time was reduced by up to 70 per cent. And, the Schur complement preconditioned FGMRES worked more efficiently than block Gauss–Seidel preconditioned FGMRES, block-selective AMG and AMG with block ILU(0) smoother and could take at least 47.4 per cent less time. The proposed solver had good scalability for different-size problems, including various cases with different numbers of elements. It also kept good speedup and efficiency in parallel performance.

A coupled solver has been proposed to effectively solve the coupled problem in the three-dimensional injection molding simulation, which is more robust and efficient than existing methods.

Discretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.

The resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.

Numerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.

The explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

The paper's aim is to devise a fast method for microwave waveguide filter permittivity tolerance analysis.

A 2D finite elements (FEs) formulation is combined via a Schur complement‐based domain decomposition (DD) technique to reduce the tolerance affected part of the analysis to a smaller domain.

The paper shows how to combine FEs and DD in an efficient way for material parameters tolerance analyses in microwave waveguide filters, showing speedup results.

The formulation here presented is 2D but can be easily extended to 3D.

The application of DD to solve numerically large problem is well‐known, the idea and organization of the algorithm to allow iteration on parameter values on a single sub‐domain is here proposed.

This paper presents new trends in parallel methods used to solve finite element matrix systems: standard iterative and direct solving methods first, and then domain decomposition methods. For example, the current status and properties of two prevailing programming environments (PVM and MPI) are finally given and compared when implemented together with a finite element time domain formulation.

External natural convection is rarely studied by numerical simulation in the literature due to the fact that flow of interest takes place in an unbounded domain and that if a limited computational domain is used the corresponding outer boundary conditions are unknown. In this study, we propose outer boundary conditions for a limited computational domain and make the corresponding numerical implementation in the scope of a projection method combining spectral methods and domain decomposition techniques. Numerical simulations are performed for both steady natural convection about an isothermal cylinder and transient natural convection around a line‐source. An experiment is also realized in water using particle image velocimetry and thermocouples to make a comparison during transients of external natural convection around a platinum wire heated by Joule effect. Good agreement, observed between numerical simulations and experiments, validated the outer boundary conditions proposed and their numerical implementation. It is also shown that, if one tolerates prediction error, numerical results obtained remain at least reasonable in a region near the line‐source during the entire transients. We thus paved the way for numerical simulation of external natural convection although further studies remain to be done for higher heating power (higher Rayleigh number).

This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.

The purpose of this paper is to simulate in the time domain three‐dimensional electrical, thermal, mechanical coupled contact problems arising in electric resistance welding (ERW) processes.

A three‐dimensional multiphysical numerical model for analyzing contact problems is proposed. Electrical and thermal field equations in bulk domains are discretized with the cell method (CM). Welding resistance at contact interfaces is described locally by synthetic statistic parameters and contacting domains are matched together by a non‐overlapping domain decomposition method. Contact pressure distribution is resolved by a finite‐element procedure. The model is validated with 3D FEM software package.

The semi‐analytical model describing the electric and thermal resistances at contact interfaces can be easily embedded in CM formulations, where problem variables are expressed directly in integral form. Compatibility conditions between contact members are enforced by a domain decomposition approach. System conditioning and computing time are improved by a solution strategy based on the Schur complement method.

The electrical‐thermal analysis is not coupled strongly with the mechanical analysis and contact pressure distribution is assumed to be not depending on thermal stresses, which can be considerable near the contact area where localized joule heating occurs.

Resistance welding processes involve mechanical, electrical, and thermal non‐linear coupled effects that cannot be simulated by standard commercial software packages. The proposed numerical model can be used instead for designing and optimizing ERW processes.

The paper shows that numerical modeling of ERW processes requires a careful prediction of the localized joule heating occurring at the electrode‐material interface. This effect is reconstructed by the proposed approach simulating coupled electrical, thermal, and mechanical effects on different spatial scales.

The purpose of this paper is to introduce the solution to two-phase flow in CO₂/brine system with salt precipitation by applying mixed hybrid finite element (MHFE) method to pressure equation and finite volume (FV) method to saturation equation. Mixed finite element method solves pressure and velocity in two subspaces while hybrid method is an extension of mixed method, where the Lagrange multiplier is added to the former in order to ensure the continuity from one element to the adjacent elements. The authors propose the modeling of salt precipitation using core flood experimental result and adapt to be applicable for numerical modeling.

The governing equations are discretized using Mixed Hybrid Finite Element-Finite Volume (MHFE-FV) method. This method has the feature of localized conservation which is attractive for application on heterogeneous porous media. In addition to this, the salt precipitation effect is modeled using the data from core flood experiment (Ott et al., 2011). The random data are linearized to obtain the relationship between salt precipitate and CO₂ saturation and implemented to the algorithm for two-phase flow in CO₂ and brine system.

The solution of MHFE-FV scheme has good agreement with the solution using implicit pressure and explicit saturation (IMPES) reported by Negara et al. (2011), with average error of 4.20 percent. Localized conservation is demonstrated in the case of randomized heterogeneous porous media where fingering effects are explicitly observed. Salt precipitation prediction using the proposed method is able to predict the decrement of porosity by 16.71 percent and permeability by 22.19 percent. This results in the decreased amount of CO₂ injected by 64.70 percent.

This paper presents the solution of two-phase flow in CO₂ brine system during CO₂ injection in saline aquifer using MHFE-FV method with the additional salt precipitation model obtained based on core flood experiment result.

A methodology to predict the salt precipitation based on CO₂ saturation.

Contribution to green house gas reduction.

The authors use MHFE-FV to solve hyperbolic PDE to obtain accurate results of CO₂ saturation, and subsequently use this to compute the salt precipitation.

Based on the framework of Krylov subspace-based model order reduction (MOR), compact models of the piezoelectric energy harvester devices can be generated. However, the stability of reduced piezoelectric model often cannot be preserved. In previous research studies, “MOR after Schur,” “Schur after MOR” and “multiphysics structure preserving MOR” methods have proven successful in obtaining stable reduced piezoelectric energy harvester models. Though the stability preservation of “MOR after Schur” and “Schur after MOR” methods has already been mathematically proven, the “multiphysics structure preserving MOR” method was not. This paper aims to provide the missing mathematical proof of “multiphysics structure preserving MOR.”

Piezoelectric energy harvesters can be represented by system of differential-algebraic equations obtained by the finite element method. According to the block structure of its system matrices, “MOR after Schur” and “Schur after MOR” both perform Schur complement transformations either before or after the MOR process. For the “multiphysics structure preserving MOR” method, the original block structure of the system matrices is preserved during MOR. 

This contribution shows that, in comparison to “MOR after Schur” and “Schur after MOR” methods, “multiphysics structure preserving MOR” method performs the Schur complement transformation implicitly, and therefore, stabilizes the reduced piezoelectric model.

The stability preservation of the reduced piezoelectric energy harvester model obtained through “multiphysics structure preserving MOR” method is proven mathematically and further validated by numerical experiments on two different piezoelectric energy harvester devices.