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A geometric multigrid (MG) method for the efficient solution of 3D non‐linear magnetostatic field problems is presented. A finite element method (FEM) with edge elements is used to describe the magnetic vector potential. A numerical example is presented to demonstrate the efficiency of the MG method not only for linear, but also for non‐linear problems.

This paper is the description of a new two‐grid algorithm to solve frictional contact problems. A regularized formulation is introduced and the discretized problem is solved using an internal non linear two‐grid technique coupled with a diagonal fixed point algorithm. Mathematical background is given, and superconvergence is obtained.

The purpose of this paper is to develop a novel unstructured simulation approach for injection molding processes described by the Hele‐Shaw model.

The scheme involves dual dynamic meshes with active and inactive cells determined from an initial background pointset. The quasi‐static pressure solution in each timestep for this evolving unstructured mesh system is approximated using a control volume finite element method formulation coupled to a corresponding modified volume of fluid method. The flow is considered to be isothermal and non‐Newtonian.

Supporting numerical tests and performance studies for polystyrene described by Carreau, Cross, Ellis and Power‐law fluid models are conducted. Results for the present method are shown to be comparable to those from other methods for both Newtonian fluid and polystyrene fluid injected in different mold geometries.

With respect to the methodology, the background pointset infers a mesh that is dynamically reconstructed here, and there are a number of efficiency issues and improvements that would be relevant to industrial applications. For instance, one can use the pointset to construct special bases and invoke a so‐called “meshless” scheme using the basis. This would require some interesting strategies to deal with the dynamic point enrichment of the moving front that could benefit from the present front treatment strategy. There are also issues related to mass conservation and fill‐time errors that might be addressed by introducing suitable projections. The general question of “rate of convergence” of these schemes requires analysis. Numerical results here suggest first‐order accuracy and are consistent with the approximations made, but theoretical results are not available yet for these methods.

This novel unstructured simulation approach involves dual meshes with active and inactive cells determined from an initial background pointset: local active dual patches are constructed “on‐the‐fly” for each “active point” to form a dynamic virtual mesh of active elements that evolves with the moving interface.

Aims to show that efficiency and accuracy of integral equation methods (IEMs) in combination with the fast multipole method for the design of a novel magnetic gear.

A novel magnetic gear was developed. Magnetic fields and torque of the gear were simulated based on IEMs. The fast multipole method was applied to compress the matrix of the belonging linear system of equations. A computer cluster was used to achieve numerical results within an acceptable time. A three‐dimensional post‐processing and visualization of magnetic fields enables a deep understanding of the gear.

IEMs are very well suited for the numerical analysis of a magnetic gear. Especially, the treatment of the air gap between the rotating components, which move with significant varying velocities, is relatively easy. Furthermore, a correct computation and visualization of flux lines is possible. A magnetic gear is advantageous for high rotational velocities.

A quasi static numerical simulation has sufficed for an understanding of the principle of the magnetic gear and for the development of a prototype.

IEMs are very suitable for the analysis of complex problems with moving parts. Nowadays, the efficiency is very good even for large problems, since matrix compression techniques are well‐engineered.

The design of a novel magnetic gear is discussed. Well‐known techniques like IEMs, fast multipole method and parallel computing are combined to solve a very large and complex problem.

The latest release of the stress analysis finite element system BERSAFE contains a new general substructuring facility for use in either elastic or non‐linear analysis. The technique allows a considerable extension to the available facilities in that parts of the structure can be stored on datafiles for current or subsequent use. In the latter case, repeated computations are avoided for components with identical geometric and material properties, so effectively larger problems can be solved without a proportional increase in cost and effort. For non‐linear analysis, the technique is well suited to cases where non‐linear behaviour is confined to certain parts of the structure, such as in the vicinity of stress concentrations and crack tips. The elastic areas can be replaced by a substructure boundary, thereby concentrating the analysis on the higher‐stressed portions of the structure and so reducing the extent and cost of each iteration. The non‐linear substructuring technique is described in detail and illustrated by examples.

The present study focusses on algorithmic aspects related to deformation dependent loads in non‐linear static finite element analysis. If the deformation dependency is considered only on the right hand side, a considerable increase in the number of iterations follows. It may also cause failure of convergence in the proximity of critical points. If in turn the deformation dependent loading is included within the consistent linearization, an additional left hand side term emerges, the so‐called load stiffness matrix. In this paper several numerical test cases are used to show and quantify the influence of the two different approaches on the iteration process. Consideration of the complete load stiffness matrix may result in a cumbersome coding effort, different for each load case, and in certain cases its derivation is even not practicable at all. Therefore also several formulations for approximated load stiffness matrices are presented. It is shown that these simplifications not only reduce the additional effort for linearization and implementation, but also keep the iterative costs relatively small and still allow the calculation of the entire equilibrium path.

The purpose of this paper is to solve non‐linear parametric thermal models defined in degenerated geometries, such as plate and shell geometries.

The work presented in this paper is based in a combination of the proper generalized decomposition (PGD) that proceeds to a separated representation of the involved fields and advanced non‐linear solvers. A particular emphasis is put on the asymptotic numerical method.

The authors demonstrate that this approach is valid for computing the solution of challenging thermal models and parametric models.

This is the first time that PGD is combined with advanced non‐linear solvers in the context of non‐linear transient parametric thermal models.

A hierarchically preconditioned conjugate gradient (PCG) method for finite element analysis is presented. Its use is demonstrated for the difficult problem of the non‐linear analysis of 3D reinforced concrete structures. Examples highlight the dramatic savings in computer storage and more modest savings in solution times obtained using PCG especially for large problems.

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 the paper is to show the application of the fixed‐point method with the T, Φ‐Φ formulation to get the steady‐state solution of the quasi‐static Maxwell's equations with non‐linear material properties and periodic excitations.

The fixed‐point method is used to solve the problem arising from the non‐linear material properties. The harmonic balance principle and a time periodic technique give the periodic solution in all non‐linear iterations. The optimal parameter of the fixed‐point method is investigated to accelerate its convergence speed.

The Galerkin equations of the DC part are found to be different from those of the higher harmonics. The optimal parameter of the fixed‐point method is determined.

The establishment of the Galerkin equations of the DC part is a new result. The method is first used to solve three‐dimensional problems with the T, Φ‐Φ formulation.