Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning ...problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.
Localization of solution of the problem of three-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based ...discrete-continual finite ele-ment method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis func-tions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.
Summary
Finite elements of class 𝒞1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment ...of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).
Finite elements of class C1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). Here we explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bezier polynomials on a structured mesh of curved quadrangular elements, which allow to remain in the physical space. Reduced cubic Hsieh‐Clough‐Tocher finite elements are then adopted on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. The Poisson problem on a two‐dimensional bounded regular domain is considered as a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio). Numerical results on the accuracy of the local interpolation and of the coupled problem are provided together with an analysis of the algorithm convergence. This analysis is a first step to the adoption of the presented methodology in the industrial code such as JOREK.
A new family of hybrid/mixed finite elements optimized for numerical stability is introduced. It comprises a linear hexahedral and quadratic hexahedral and tetrahedral elements. The element ...formulation is derived from a consistent linearization of a well-known three-field functional and related to Simo–Taylor–Pister (STP) elements. For the quadratic hexahedral and tetrahedral elements we derive (static reduced) discontinuous hybrid elements, as well as continuous mixed finite elements with additional primary unknowns for the hydrostatic pressure and the dilation, whereas the linear hexahedral element is of the discontinuous type. The elements can readily be used in combination with any isotropic, invariant-based hyperelastic material model and can be considered as being locking-free. In a representative numerical benchmark test the elements numerical stability is assessed and compared to STP-elements and the family of discontinuous hybrid elements implemented in the commercial finite element code Abaqus/Standard. The new elements show a significant advantage concerning the numerical robustness.
This paper focuses on the design and analysis of a novel material-efficient permanent-magnet (PM) shape for surface-mounted PM (SPM) motors used in automotive actuators. Most of such applications ...require smooth torque with minimum pulsation for an accurate position control. The proposed PM shape is designed to be sinusoidal and symmetrical in the axial direction for minimizing the amount of rare earth magnets as well as for providing balanced axial electromagnetic force, which turns out to obtain better sinusoidal electromotive force, less cogging torque, and, consequently, smooth electromagnetic torque. The contribution of the novel PM shape to motor characteristics is first estimated by 3-D finite-element method, and all of the simulation results are compared with those of SPM motors with two conventional arched PM shapes: one previously reported sinusoidal PM shape and one step skewed PM shape. Finally, some finite-element analysis results are confirmed by experimental results.
This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main ...comparison result is that the error of the P2P0-FEM is a lower bound to the error of the Bernardi–Raugel (or reduced P2P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs.
Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.
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