Phase-field modeling of dendritic growth under forced flow based on adaptive finite element method

A mathematical model combined projection algorithm with phase-field method was applied. The adaptive finite element method was adopted to solve the model based on the non-uniform grid, and the behavior of dendritic growth was simulated from undercooled nickel melt under the forced flow. The simulation results show that the asymmetry behavior of the dendritic growth is caused by the forced flow. When the flow velocity is less than the critical value, the asymmetry of dendrite is little influenced by the forced flow. Once the flow velocity reaches or exceeds the critical value, the controlling factor of dendrite growth gradually changes from thermal diffusion to convection. With the increase of the flow velocity, the deflection angle towards upstream direction of the primary dendrite stem becomes larger. The effect of the dendrite growth on the flow field of the melt is apparent. With the increase of the dendrite size, the vortex is present in the downstream regions, and the vortex region is gradually enlarged. Dendrite tips appear to remelt. In addition, the adaptive finite element method can reduce CPU running time by one order of magnitude compared with uniform grid method, and the speed-up ratio is proportional to the size of computational domain.

Adaptive Finite Element Method Based on Optimal Error Estimates for Linear Elliptic Problems on Nonconvex Polygonal Domains

The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.

ADAPTIVE FINITE ELEMENT METHOD BASED ON OPTIMAL ERROR ESTIMATES FOR LINEAR ELLIPTIC PROBLEMS ON CONCAVE CORNER DOMAINS

An adaptive finite element procedure designed for specific computational goals is presented,using mesh refinement strategies based on optimal or nearly optimal a priori error estimates for the finite element method and using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions.The proposed procedure is analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that the method can generate the correct type of refinements and lead to the desired control under consideration.

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.

An Adaptive Finite Element Method Based on Optimal Error Estimates for Linear Elliptic Problems

The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.

Adaptive hp finite element method for fluorescence molecular tomography with simplified spherical harmonics approximation

Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.

Adaptive Finite Element Method Based on Optimal Error Estimtes for Linear Elliptic Problems on Concave Corner Domains (continuation)

This paper is the third part in a series of papers on adaptive finite elementmethods based on optimal error estimates for linear elliptic problems on the concavecorner domains. In this paper, a result is obtained. The algorithms for error controlboth in the energy norm and in the maximum norm presented in part 1 and part 2 ofthis series arc based on this result.

WEAK-DUALITY BASED ADAPTIVE FINITE ELEMENT METHODS FOR PDE-CONSTRAINED OPTIMIZATION WITH POINTWISE GRADIENT STATE-CONSTRAINTS

Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error estimators.Mathematics subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.

An Adaptive Finite Element Method with a Modified Perfectly Matched Layer Formulation for Diffraction Gratings

For numerical simulation of one-dimensional diffraction gratings both in TE and TM polarization,an enhanced adaptive finite element method is proposed in this paper.A modified perfectly matched layer(PML)formulation is proposed for the truncation of the unbounded domain,which results in a homogeneous Dirichlet boundary condition and the corresponding error estimate is greatly simplified.The a posteriori error estimates for the adaptive finite element method are provided.Moreover,a lower bound is obtained to demonstrate that the error estimates obtained are sharp.

Design of Finite Element Tools for Coupled Surface and Volume Meshes

Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate'matching'local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.

METHOD FOR ADAPTIVE MESH GENERATION BASED ON GEOMETRICAL FEATURES OF 3D SOLID

In order to provide a guidance to specify the element size dynamically during adaptive finite element mesh generation, adaptive criteria are firstly defined according to the relationships between the geometrical features and the elements of 3D solid. Various modes based on different datum geometrical elements, such as vertex, curve, surface, and so on, are then designed for generating local refined mesh. With the guidance of the defmed criteria, different modes are automatically selected to apply on the appropriate datum objects to program the element size in the local special areas. As a result, the control information of element size is successfully programmed covering the entire domain based on the geometrical features of 3D solid. A new algorithm based on Delatmay triangulation is then developed for generating 3D adaptive finite element mesh, in which the element size is dynamically specified to catch the geometrical features and suitable tetrahedron facets are selected to locate interior nodes continuously. As a result, adaptive mesh with good-quality elements is generated. Examples show that the proposed method can be successfully applied to adaptive finite element mesh automatic generation based on the geometrical features of 3D solid.

THE ADAPTIVE IMMERSED INTERFACE FINITE ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS

In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λ- independent multiplicative constants; the Lame constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.

An Adaptive Finite Element PML Method for the Acoustic Scattering Problems in Layered Media

The paper concerns the numerical solution for the acoustic scattering problems in a two-layer medium.The perfectly matched layer(PML)technique is adopted to truncate the unbounded physical domain into a bounded computational domain.An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem.Numerical experiments are included to demonstrate the efficiency of the proposed method.

A LOCAL MULTILEVEL PRECONDITIONER FOR THE ADAPTIVE MORTAR FINITE ELEMENT METHOD＊

In this paper, we propose a local multilevel preconditioner for the mortar finite element approximations of the elliptic problems. With some mesh assumptions on the interface, we prove that the condition number of the preconditioned systems is independent of the large jump of the coefficients but depends on the mesh levels around the cross points. Some numericM experiments are presented to confirm our theoreticM results.

Numerical Solution of Acoustic Scattering by an Adaptive DtN Finite Element Method

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions,where the wave propagation is governed by the Helmholtz equation.The scattering problem is modeled as a boundary value problem over a bounded domain.Based on the Dirichlet-to-Neumann(DtN)operator,a transparent boundary condition is introduced on an artificial circular boundary enclosing the obstacle.An adaptive finite element based on a posterior error estimate is presented to solve the boundary value problem with a nonlocal DtN boundary condition.Numerical experiments are included to compare with the perfectly matched layer(PML)method to illustrate the competitive behavior of the proposed adaptive method.

ERROR REDUCTION, CONVERGENCE AND OPTIMALITY FOR ADAPTIVE MIXED FINITE ELEMENT METHODS FOR DIFFUSION EQUATIONS

Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods （AMFEM） for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.

OPTIMALITY OF LOCAL MULTILEVEL METHODS FOR ADAPTIVE NONCONFORMING P1 FINITE ELEMENT METHODS

In this paper, a local multilevel product algorithm and its additive version are con- sidered for linear systems arising from adaptive nonconforming P1 finite element approx- imations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jaeobi or Gauss-Seidel smoothers per- formed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

Characteristics of permanent magnet linear synchronous motor fed by spwm inverter based on field-circuit coupled method

Presented field-circuit coupled adaptive time-stepping finite element method to study on permanent magnet linear synchronous motor (PMLSM) characteristics fed by SPWM voltage source inverter.In air-gap field where the direction or magnitude of the field is changing rapidly,the smallest elements are demanded due to high accuracy to use adaptive meshing technique.The co-simulation was used with the status space functions and time-step finite element functions,in which time-step of the status space functions was the smallest than finite element functions'.The magnitude relation of the normal elec- tromagnetic force and tangential electromagnetic force and the period were attained,and current curve was very abrupt at current zero area due to the bigger resistance and leak- age reactance,including main characteristics of motor voltage and velocity.The simulation results compare triumphantly with the experiments results.

Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.

Multi-mesh Adaptive Finite Element Algorithms for Constrained Optimal Control Problems Governed By Semi-Linear Parabolic Equations

In this paper, we derive a posteriori error estimators for the constrained optimal control problems governed by semi-linear parabolic equations under some assumptions. Then we use them to construct reliable and efficient multi-mesh adaptive finite element algorithms for the optimal control problems. Some numerical experiments are presented to illustrate the theoretical results.