Parallel anisotropic mesh adaptivity with dynamic load balancing for cardiac electrophysiology

Simulations in cardiac electrophysiology generally use very fine meshes and small time steps to resolve highly localized wavefronts. This expense motivates the use of mesh adaptivity, which has been demonstrated to reduce the overall computational load. However, even with mesh adaptivity performing such simulations on a single processor is infeasible. Therefore, the adaptivity algorithm must be parallelised. Rather than modifying the sequential adaptive algorithm, the parallel mesh adaptivity method introduced in this paper focuses on dynamic load balancing in response to the local refinement and coarsening of the mesh. In essence, the mesh partition boundary is perturbed away from mesh regions of high relative error, while also balancing the computational load across processes. The parallel scaling of the method when applied to physiologically realistic heart meshes is shown to be good as long as there are enough mesh nodes to distribute over the available parallel processes. It is shown that the new method is dominated by the cost of the sequential adaptive mesh procedure and that the parallel overhead of inter-process data migration represents only a small fraction of the overall cost.     

Adaptive finite element analysis of structures under transient dynamic loading using modal superposition

In this paper, the error estimation and adaptive strategy developed for the linear elastodynamic problem under transient dynamic loading based on the Z–Z criterion is utilized for 2D and plate bending problems. An automatic mesh generator based on “growth meshing” is utilized effectively for adaptive mesh refinement. Optimal meshes are obtained iteratively corresponding to the prescribed domain discretization error limit and for a chosen number of basis modes satisfying modal truncation errors. Numerous examples show the effectiveness of the integrated approach in achieving the target accuracy in finite element transient dynamic analysis.     

Material-Force-Based Refinement Indicators in Adaptive Strategies

A material-force-based refinement indicator for adaptive finite element strategies for finite elasto-plasticity is proposed. Starting from the local format of the spatial balance of linear momentum, a dual material counterpart in terms of Eshelby's energy-momentum tensor is derived. For inelastic problems, this material balance law depends on the material gradient of the internal variables. In a global format the material balance equation coincides with an equilibrium condition of material forces. For a homogeneous body, this condition corresponds to vanishing discrete material nodal forces. However, due to insufficient discretization, spurious material forces occur at the interior nodes of the finite element mesh. These nodal forces are used as an indicator for mesh refinement. Assigning the ideas of elasticity, where material forces have a clear energetic meaning, the magnitude of the discrete nodal forces is used to define a relative global criterion governing the decision on mesh refinement. Following the same reasoning, in a second step a criterion on the element level is computed which governs the local h-adaptive refinement procedure. The mesh refinement is documented for a representative numerical example of finite elasto-plasticity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints

In this paper, we study a posteriori error estimates of the upwind symmetric interior penalty Galerkin (SIPG) method for the control constrained optimal control problems governed by linear diffusion–convection–reaction partial differential equations. Residual based error estimators are used for the state, the adjoint and the control. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented for convection dominated problems to illustrate the theoretical findings and the effectiveness of the adaptivity.     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sobre la necesidad de controlar el error de discretización de elementos finitos en optimización de forma estructural con algoritmos evolutivos

This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of the algorithm described by Bugeda et al. [1] which reduces the computational cost associated to the adaptive FE analysis of each geometrical configuration and, (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.     

Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems

We study potential advantages of adaptive mesh point selection for the solution of systems of initial value problems. For an optimal order discretization method, we propose an algorithm for successive selection of the mesh points, which only requires evaluations of the right-hand side function. The selection (asymptotically) guarantees that the maximum local error of the method does not exceed a prescribed level. The usage of the algorithm is not restricted to the chosen method; it can also be applied with any method from a general class. We provide a rigorous analysis of the cost of the proposed algorithm. It is shown that the cost is almost minimal, up to absolute constants, among all mesh selection algorithms. For illustration, we specify the advantage of the adaptive mesh over the uniform one. Efficiency of the adaptive algorithm results from automatic adjustment of the successive mesh points to the local behavior of the solution. Some numerical results illustrating theoretical findings are reported.     

Comparison of refinement criteria for structured adaptive mesh refinement

We have investigated and analyzed the grid convergence issues for an adaptive mesh refinement (AMR) code. We have found that the numerical results for the AMR grid may have a larger error than those for the unrefined uniform grid. After a detailed analysis, we have found that the numerical solution at the coarse-fine interface between different levels of the grid converges only in the first-order accuracy. Therefore, the error near the coarse-fine interface can quickly dominate the error in the other regions if the coarse-fine interface is active and not covered by the fine grid. We propose, implement, and compare several refinement criteria. Some of them can catch the large-error region near the coarse-fine interface and refine them with the fine grid.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

Towards a more efficient dynamic mesh adaptation methodology for continuum discretization in complex engineering problems

A novel and efficient method of adaptive mesh generation, for dynamically adaptive unstructured grids, is proposed. A locally refined triangulation is constructed on a coarse background mesh, subdividing each triangle in the refinement region R into four congruent sub-triangles iteratively, by connecting edge midpoints, until triangles of a prescribed lengthscale are obtained. The unavoidable propagation outside the refinement region R is restricted to a single triangle in the coarse background mesh. The triangles, in the immediate vicinity of region R, are broken down using the concept of iterated function systems, widely used in fractal modeling, by recursive generation of sub-triangles with a gradation towards the region R triangles. A quantitative assessment of the present algorithm proves its superiority over other comparable models reported in the literature. The time cost of the algorithm is linear, and the method can be easily extended to three dimensions.     

Adaptive refinement procedure for the finite difference method

An adaptive refinement procedure consisting of a localized error estimator and a physically based approach to mesh refinement is developed for the finite difference method. The error estimator is a variation of a successful finite element error estimator. The errors are estimated by computing an error energy norm in terms of discontinuous and continuous stress fields formed from the finite difference results for plane stress problems. The error measure identifies regions of high error which are subsequently refined to improve the result. The local refinement procedure utilizes a recently developed approach for developing finite difference templates to produce a graduated mesh. The adaptive refinement procedure is demonstrated with a problem that contains a well-defined singularity. The results are compared to finite element and uniformly refined finite difference results.     

Simulating cardiac electrophysiology using anisotropic mesh adaptivity

The simulation of cardiac electrophysiology requires small time steps and a fine mesh in order to resolve very sharp, but highly localized, wavefronts. The use of very high resolution meshes containing large numbers of nodes results in a high computational cost, both in terms of CPU hours and memory footprint. In this paper an anisotropic mesh adaptivity technique is implemented in the Chaste physiological simulation library in order to reduce the mesh resolution away from the depolarization front. Adapting the mesh results in a reduction in the number of degrees of freedom of the system to be solved by an order of magnitude during propagation and 2–3 orders of magnitude in the subsequent plateau phase. As a result, a computational speedup by a factor of between 5 and 12 has been obtained with no loss of accuracy, both in a slab-like geometry and for a realistic heart mesh with a spatial resolution of 0.125 mm.     

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

This work combines two complementary strategies for solving the steady incompressible Navier–Stokes model with a zeroth‐order term, namely, a stabilized finite element method and a mesh–refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth‐order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Four-triangles adaptive algorithms for RTIN terrain meshes

We present a refinement and coarsening algorithm for the adaptive representation of Right-Triangulated Irregular Network (RTIN) meshes. The refinement algorithm is very simple and proceeds uniformly or locally in triangle meshes. The coarsening algorithm decreases mesh complexity by reducing unnecessary data points in the mesh after a given error criterion is applied. We describe the most important features of the algorithms and give a brief numerical study on the propagation associated with the adaptive scheme used for the refinement algorithm. We also present a comparison with a commercial tool for mesh simplification, Rational Reducer, showing that our coarsening algorithm offers better results in terms of accuracy of the generated meshes.     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.     

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

Adaptive finite element methods for the identification of distributed parameters in elliptic equation

In this paper, adaptive finite element method is developed for the estimation of distributed parameter in elliptic equation. Both upper and lower error bound are derived and used to improve the accuracy by appropriate mesh refinement. An efficient preconditioned project gradient algorithm is employed to solve the nonlinear least-squares problem arising in the context of parameter identification problem. The efficiency of our error estimators is demonstrated by some numerical experiments.      

ADAPTIVITY IN SPACE AND TIME FOR MAGNETOQUASISTATICS

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell＇s equations on unstructured 3D tetrahedral grids. Spatial mesh re- finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H（curl）-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.