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This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h‐adaptive finite element method is adopted for spatial discretization and implemented with matrix‐free operator evaluation. The ground state energy is found by performing unconstrained minimization with non‐orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward O(N) complexity for the locally adaptive real‐space solution of density functional theory with finite elements. [ABSTRACT FROM AUTHOR]

Freeze‐casting has attracted great attention as a potential method for manufacturing bioinspired materials with excellent flexibility in microstructure control. The solidification of ice crystals in ceramic colloidal suspensions plays an important role during the dynamic process of freeze‐casting. During solidification, the formation of a microstructure results in a dendritic pattern within the ice‐template crystals, which determines the macroscopic properties of materials. In this paper, the authors propose a phase‐field model that describes the crystallization in an ice template and the evolution of particles during anisotropic solidification. Under the assumption that ceramic particles represent mass flow, namely a concentration field, the authors derive a sharp‐interface model and then transform the model into a continuous initial boundary value problem via the phase‐field method. The adaptive finite‐element technique and generalized single‐step single‐solve (GSSSS) time‐integration method are employed to reduce computational cost and reconstruct microstructure details. The numerical results are compared with experimental results, which demonstrate good agreement. Finally, a microstructural morphology map is constructed to demonstrate the effect of different concentration fields and input cooling rates. The authors observe that at particle concentrations ranging between 25 and 30% and cooling rate lower than −5° min−1 generates the optimal dendrite structure in freeze casting process. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we present a finite element method with a residual-based artificial viscosity for simulation of turbulent compressible flow, with adaptive mesh refinement based on a posteriori error estimation with sensitivity information from an associated dual problem. The artificial viscosity acts as a numerical stabilization, as shock capturing, and as turbulence capturing for large eddy simulation of turbulent flow. The adaptive method resolves parts of the flow indicated by the a posteriori error estimates but leaves shocks and turbulence under-resolved in a large eddy simulation. The method is tested for examples in 2D and 3D and is validated against experimental data. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we present the development of interior magnet motors reducing iron loss at high rotational speed under the flux weakening control. The rotor core and magnet shapes are determined by the automatic numerical calculation using combination of the optimization method and the adaptive finite element method. The optimized motor is manufactured to prove the effectiveness by the measurement of the iron loss. Both results of the calculation and the measurement indicate that the iron loss of the proposed motor at the high rotational speed under the flux weakening control is reduced by half compared with the initial rotor shape while the torque is nearly constant. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(3): 59–66, 2009; Published online in Wiley InterScience (www.interscience. wiley.com). DOI 10.1002/eej.20827 [ABSTRACT FROM AUTHOR]

In this paper, details of an implementation of a numerical code for computing the Kohn-Sham equations are presented and discussed. A fully self-consistent method of solving the quantum many-body problem within the context of density functional theory using a real-space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co-aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon-monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. At the heart of the solver is a goal‐oriented a posteriori error estimate for the electrostatic coupling, derived and implemented in the present work, that gives rise to an orders of magnitude improved precision and a shorter computational time as compared to standard finite difference solvers. The accuracy of the new solver ARGOS is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths and between the spike protein of SARS‐CoV‐2 and the ACE2 receptor. All the calculations are repeated by using the well‐known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver. [ABSTRACT FROM AUTHOR]

This study presents a new framework to solve the coupled semilinear elliptic equation by the domain decomposition algorithm. Unlike the traditional domain decomposition algorithm, the coupled semilinear elliptic equation doesn't need to be solved directly. The strategy is to construct a set of nested finite element spaces, and subsequently solve some decoupled linear elliptic equations by using the domain decomposition method in each level space. Additionally, a small‐scale coupled semilinear elliptic equation in a specially designed correction space will be solved. As the large‐scale coupled semilinear elliptic equation doesn't need to be solved directly, there will be an improved efficiency as compared to the traditional domain decomposition method. Furthermore, as the domain decomposition method is only used to solve decoupled linear elliptic equations, any efficient algorithms designed for the associated linear elliptic equations can be incorporated in the proposed algorithm framework. Thus, the algorithm is highly flexible. Additionally, it can be theoretically proven that the proposed algorithm has very low requirements for the smoothness of nonlinear terms. [ABSTRACT FROM AUTHOR]

In the paper the authors investigate the conditions of optimal mesh, and prove that the element errors in the optimal mesh should be identical for all existing elements. Also, it is shown that the correct mesh refinement formulations should be derived at an element level rather than the global level. Thus the element permissible errors need to be determined. The method to determine these permissible element errors is described. [ABSTRACT FROM AUTHOR]

Summary: Homogenization methods are drawing increasing attention for simulation of heterogeneous materials like composites. For balancing the accuracy and the numerical efficiency of such strategies, we deal with both model and discretization errors of the finite element method (FEM) on a macroscale. Within a framework of goal‐oriented adaptivity, we consider linear elastic heterogeneous materials, for which first‐order homogenization schemes apply. A novel model hierarchy is proposed based on mean‐field and full‐field homogenization methods. For the former, we consider several well‐established schemes like Mori‐Tanaka or self‐consistent as basic models, and for the latter, as superior models, unit cell problems are solved via the FEM under an a priori chosen boundary condition. For a further stage of the model hierarchy, we consider hierarchical unit cells within the frame of the FEM toward an adaptive selection of the unit cell size. By means of several numerical examples, we illustrate the effectiveness of the proposed adaptive approach. [ABSTRACT FROM AUTHOR]

In this paper, we address a 3D fluid-structure interaction benchmark problem that represents important characteristics of biomedical modeling. We present a goal-oriented adaptive finite element methodology for incompressible fluid-structure interaction based on a streamline diffusion-type stabilization of the balance equations for mass and momentum for the entire continuum in the domain, which is implemented in the Unicorn/FEniCS software framework. A phase marker function and its corresponding transport equation are introduced to select the constitutive law, where the mesh tracks the discontinuous fluid-structure interface. This results in a unified simulation method for fluids and structures. We present detailed results for the benchmark problem compared with experiments, together with a mesh convergence study. [ABSTRACT FROM AUTHOR]

In this article new contributions to the theory and computation of cyclic martensitic phase transformations (PT) in mono- and poly-crystalline metallic shape memory alloys are presented. The PT models of the non-convex variational problem are based on the Cauchy-Born hypothesis and Bain's principle. A quasi-convexified C1-continuous thermo-mechanical micro-macro constitutive model for metallic monocrystals is developed which is represented together with the phase transformation constraints by a unified Lagrangian variational functional including phase evolution equations with mass conservation. The unified setting presented here includes poly-crystalline shape memory alloys whose microstructure is modeled using lattice variants. A pre-averaging scheme for randomly distributed poly-crystalline variants of transformation strains is used to transform them into those of a fictitious monocrystal. Thus, the incremental integration in process time and the spatial integration algorithms of the discrete variational problems for both mono- and poly-crystalline phase transformations can be implemented into a unified algorithm with branching for mono- and poly-crystalline phase transformations. Furthermore, an error-controlled adaptive 3D finite element method in space is presented for phase transformation problems using explicit error indicator with gradient smoothing and mesh refinements via new mesh generation in each adaptive step. Computations of informative examples with convergence studies, and comparisons with published experimental results are presented using 3D finite elements. [ABSTRACT FROM AUTHOR]

A method for the modification of finite element meshes based on adaptive meshing for optimization of electric machines is proposed. In the optimization of the machine, the proposed method generates the mesh for the new shape from the previous mesh with minor modification. The locations of the nodes for the new shape are decided by solving a Laplacian equation whose unknowns are the displacements of the nodes. The advantages are illustrated by application to the shape optimization of an IPM motor. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(1): 21–28, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20773 [ABSTRACT FROM AUTHOR]