1,721,198 research outputs found
Matrix properties of a vector potential cell method for magnetostatics
No full textIn this paper, a proof of the symmetry of the system matrix arising in a class of vector potential cell methods with non-symmetric material matrices is presented. Some remarks on how to construct other schemes with symmetric system matrices are also presented. The explicit expression of the matrix entries derived in order to prove the symmetry is also used to show that this matrix is identical to the one arising in the standard edge finite-element method. Finally, some remarks on discrete regularizations of these formulations are given
Convergence analysis of Padé approximations for Helmholtz frequency response problems
Full text linkFast least-squares Padé approximation of problems with normal operators and meromorphic structure
Full text linkIn this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g., the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Pade approximation technique studied in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284] following [J. Approx. Theory 95 (1998), pp. 203-2124]. In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the "fast" in the name). Moreover, we prove several theoretical results that improve and extend those in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284], including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Pade approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in [ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284] for differential operators with normal and compact resolvent
An adaptive mixed formulation for 3D magnetostatics
No full textAn enhanced version of a mixed field-based formulation for magnetostatics previously developed by the authors is presented and its features are discussed. The formulation minimises the residual of the constitutive equation, and exactly imposes Maxwell’s equations with Lagrange multipliers. Finite elements satisfying the physical continuity properties for both the magnetic and the magnetic induction fields are used in the numerical approximation. The possibility of decoupling the formulation in two separate sets of equations is discussed. A preconditioned iterative method to solve the final algebraic linear system is presented. Finally, a very natural refinement indicator is defined to guide an adaptive mesh refinement procedure
Mixed finite element methods and tree-cotree implicit condensation
No full textMixed methods are widely used for the finite element analysis of differential problems. From a computational point of view, however, the increase in the number of variables from that of the original problem and the properties of the resulting matrices often make a direct implementation of these formulations inadvisable. In this paper, the mixed formulation of Dirichlet’s problem for the Laplace operator, discretised using lowest degree Raviart–Thomas elements will be considered as an example. A tree-cotree decomposition of the graph associated with the mesh allows the implemen- tation of an equivalent method, which significantly reduces the number of degrees of freedom and gives rise to a symmetric positive definite linear system. We will refer to this novel method as tcic (Tree-Cotree Implicit Condensation). Numerical results will be presented, and the possibility of extending the proposed method to other cases will be discussed
A field-based finite element method for magnetostatics derived from an error minimization approach
No full textAn enhanced version of a mixed field-based formulation for magnetostatics developed in previous papers is presented under more general hypotheses and a deep discussion of its features is carried out. The approach relies upon the minimization of the residual of the constitutive equation under constraints represented by Maxwell's equations, which are exactly imposed with Lagrange multipliers. In order to obtain computed fields with correct continuity properties across interfaces between materials with different permeability, the magnetic and the magnetic induction fields are used in a complementary way, and discretized by edge and face elements. Moreover, it is discussed how the formulation can be decomposed into two separate sets of equations, highlighting the relationship with classical formulations. A preconditioned iterative scheme to solve the final algebraic linear system is also presented. Furthermore, a very natural refinement indicator is defined to guide an adaptive mesh refinement procedure
Discontinuous finite element methods for the simulation of rotating electrical machines
No full textThe capability of discontinuous finite element methods of handling non-matching grids is exploited in the simulation of rotating electrical machines. During time stepping, the relative movement of two meshes, consistent with two different regions of the electrical device (rotor and stator), results in the generation of so-called hanging nodes on the slip surface. A discretisation of the problem in the air-gap region between rotor and stator, which relies entirely on finite element methods, is presented here. A discontinuous Galerkin method is applied in a small region containing the slip surface, and a conforming method is used in the remaining part
Tree-cotree implicit condensation in magnetostatics
No full textA tree-cotree decomposition of the graph associated with a finite-element mesh allows the implementation of a method which is equivalent to a mixed one but which gives rise to a symmetric positive definite linear system with a reduced number of unknowns. We will refer to this method as TCIC (Tree-Cotree Implicit Condensation). A possible preconditioning scheme for the resulting linear system is presented and the effect of the choice of particular tree-cotree decompositions is discussed from the point of view of numerical performanc
A structure-preserving discontinuous Galerkin scheme for the Fisher–KPP equation
Full text linkAn implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L-1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L-2 norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results
Efficient use of the local discontinuous Galerkin method for meshes sliding on a circular boundary
No full textIn this paper, the coupling of discontinuous finite elements (FEs) with standard conforming ones is applied to the special case of rotating electrical machines. The proposed scheme exploits the capability of discontinuous methods of dealing with non-matching grids, and the lower computational cost of conforming methods, by using first ones only where needed. Therefore, the technique is ideally suited for the treatment of the air-gap region of such devices where the rotation of one part of the mesh generates hanging nodes. The resulting purely finite element scheme is applied to the TEAM 24 benchmark proble
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