Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications

In this paper, an object-oriented framework for numerical analysis of multi-physics applications is presented. The framework is divided into several basic sets of classes that enable the code segments to be built according to the type of problem to be solved. Fortran 2003 was used in the development of this finite element program due to its advantages for scientific and engineering programming and its new object-oriented features. The program was developed with h-type adaptive mesh refinement, and it was tested for several classical cases involving heat transfer, fluid mechanics and structural mechanics. The test cases show that the adaptive mesh is refined only in the localization region where the feature gradient is relatively high. The overall mesh refinement and the h-adaptive mesh refinement were justified with respect to the computational accuracy and the CPU time cost. Both methods can improve the computational accuracy with the refinement of mesh. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.     

卫星导航多干扰源直接定位方法

卫星导航信号到达地面时非常微弱,容易受到各种干扰,给用户带来一定的影响;针对此种情况,面向卫星导航系统多干扰源定位场景,传统两步定位算法受参数估计精度影响较大, DPD_MVDR算法虽然改进了MUSIC算法需要估计目标数这一缺点,但由于使用固定网格分辨率,定位精度与计算复杂度二者无法兼得;针对上述问题,提出一种改进DPD_MVDR的直接定位方法,在直接定位这一关键技术上运用自适应网格细化,其最大优势在于只对干扰源位置附近的网格实现多级细化,能很好地兼顾定位精度与计算复杂度,避免了传统穷举搜索带来的巨大计算复杂度。首先使用DPD_MVDR算法在粗网格下进行位置初始估计,然后在估计位置处进行迭代自适应网格细化,在降低计算复杂度的同时,提高定位精度。仿真表明,改进算法在100 m网格分辨率下计算复杂度明显降低且较DPD_MVDR算法定位精度得到明显提高;适用于对定位精度和定位实时性均有一定要求的场景。     

A posteriori error estimation and adaptive mesh refinement for combined thermal-stress finite element analysis

Local and global error estimators and an associated h-based adaptive mesh refinement schemes are proposed for coupled thermal-stress problems. The error estimators are based on the “flux smoothing” technique of Zienkiewicz and Zhu with important modifications to improve convergence performance and computational efficiency. Adaptive mesh refinement is based on the concept of adaptive accuracy criteria, previously presented by the authors for stress-based problems and extended here for coupled thermal-stress problems. Three methods of mesh refinement are presented and numerical results indicate that the proposed method is the most efficient in terms of number of adaptive mesh refinements required for convergence in both the thermal and stress solutions. Also, the proposed method required a smaller number of active degrees of freedom to obtain an accurate solution.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

High-order,multidimensional, and conservative coarse–fine interpolation for adaptive mesh refinement

The author presents a polynomial-based algorithm for high-order multidimensional interpolation at the coarse–fine interface in the context of adaptive mesh refinement on structured Cartesian grids. The proposed algorithm reduces coarse–fine interpolation to matrix–vector products by exploiting the static mesh geometry and a family of nonsingularity-preserving stencil transformations. As such, no linear system is solved at the runtime and the ill-conditioning of Vandermonde matrix is avoided. The algorithm is also generic in that D, the dimensionality of the computational domain, and p, the degree of the interpolating polynomial, are both arbitrary positive integers. Stability and accuracy are verified by interpolating simple functions, and by applying the proposed method to adaptively solving Poisson’s equation and the convection–diffusion equation. The companion MATLAB® package, AMRCFI, is also freely available for convenience and more implementation details.     

An efficient block parallel AMR method for two phase interfacial flow simulations

The direct numerical simulation of two phase interfacial flows can be computationally challenging, as the strong resolution needed to follow the deformations of the interface leads to a lot of time spent solving the whole computation domain. Efficient solution of such problems requires an adaptive mesh refinement capability to concentrate computational effort where it is most needed. In this paper a parallel adaptive algorithm to solve incompressible two-phase flows with surface tension is presented: the AMR is handled with the help of the PARAMESH package. The free interface between fluids is tracked via Level Set approach; the jump conditions at the interface for pressure and velocity are imposed by the Ghost Fluid method. A multigrid preconditioned BiCG-stab solver adapted to the AMR data structure has been developed to allow high density ratio computations (up to 1:1000). Special treatment has been done at the refinement jumps to maintain the fine mesh accuracy. Computational results are compared in different test cases with analytical solutions or literature, and show very good agreement with the references. The effectiveness of PARAMESH parallelization has been quite well maintained, as shown in the strong and weak scaling tests. Speed-up capabilities of the AMR are demonstrated.     

An application-centric characterization of domain-based SFC partitioners for parallel SAMR

Structured adaptive mesh refinement (SAMR) methods for the numerical solution of partial differential equations yield highly advantageous ratios for cost/accuracy as compared to methods based on static uniform approximations. These techniques are being effectively used in many domains including computational fluid dynamics, numerical relativity, astrophysics, subsurface modeling, and oil reservoir simulation. Distributed implementations of these methods, however, lead to significant challenges in dynamic data-distribution, load-balancing, and runtime management. This paper presents an application-centric characterization of a suite of dynamic domain-based inverse space-filling curve partitioning techniques for the distributed adaptive grid hierarchies that underlie SAMR applications. The overall goal of this research is to formulate policies required to drive a dynamically adaptive metapartitioner for SAMR grid hierarchies capable of selecting the most appropriate partitioning strategy at runtime based on current application and system state. Such a metapartitioner can significantly reduce the execution time of SAMR applications.     

Adaptive mesh refinement for shells with modified Ahmad elements

This paper presents a simple scheme for the generation of a quadrilateral element mesh for shells with arbitrary three-dimensional geometry. The present mesh generation scheme incorporates a normal mesh generator for generating a mesh in the two-dimensional plane and a specific mapping technique which maps the two-dimensional mesh onto the three-dimensional curved surface. As the mapping is a one-to-one mapping between the mesh in the plane and that on the curved surface, the resulting surface discretization is compatible with the local mesh parameters in two dimensions. This scheme is further combined, both with a sophisticated error estimate determined by using the best guess values of bending moments and membrane and transverse shear forces obtained from a previous solution, and an effective mesh refinement strategy established at an element level in order to complete an adaptive analysis for shell structures. Numerical examples are shown to illustrate the principles and procedure of the present adaptive analysis.     

An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.     

Controlled cost of adaptive mesh refinement in practical 3D finite element analysis

In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper we propose the use of alternative mesh refinement with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria allow to obtain the maximum of accuracy for a prescribed cost. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.     

An optimal constrained approach for divergence-free velocity interpolation and multilevel VOF method

The use of mesh refinement techniques is becoming more and more popular in computational fluid dynamics, from multilevel approaches to adaptive mesh refinement. In this paper we present a new method to interpolate the coarse velocity field which is based on an optimal approach and is characterized by a constrained minimization of an objective functional. The functional contains the sum of the square difference between the velocity components and their target average value subject to a number of divergence-free constraints. In this work we describe this approach in two- and three-dimensional geometries with different discrete velocity field configurations. This technique is applied to a multilevel Volume-of-Fluid (VOF) method where the volume fraction function is used to reconstruct and advect the interface between two immiscible phases. The coarse velocity field is interpolated to a fixed fine grid with the optimal approach over a given number of refinement levels. The results of several kinematic tests are presented, where the mass and geometrical errors are compared with those obtained with refined velocity fields interpolated with a simple midpoint rule.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

SUT-DAM: An integrated software environment for multi-disciplinary geotechnical engineering

As computer simulation increasingly supports engineering design, the requirement for a computer software environment providing an integration platform for computational engineering software increases. A key component of an integrated environment is the use of computational engineering to assist and support solutions for complex design. In the present paper, an integrated software environment is demonstrated for multi-disciplinary computational modeling of structural and geotechnical problems. The SUT-DAM is designed in both popularity and functionality with the development of user-friendly pre- and post-processing software. Pre-processing software is used to create the model, generate an appropriate finite element grid, apply the appropriate boundary conditions, and view the total model. Post-processing provides visualization of the computed results. In SUT-DAM, a numerical model is developed based on a Lagrangian finite element formulation for large deformation dynamic analysis of saturated and unsaturated soils. An adaptive FEM strategy is used into the large displacement finite element formulation by employing an error estimator, adaptive mesh refinement, and data transfer operator. This consists in defining new appropriate finite element mesh within the updated, deformed geometry and interpolating (mapping) the pertinent variables from one mesh to another in order to continue the analysis. The SUT-DAM supports different yield criteria, including classical and advanced constitutive models, such as the Pastor–Zienkiewicz and cap plasticity models. The paper presents details of the environment and includes several examples of the integration of application software.     

A multidimensional grid-adaptive relativistic magnetofluid code

A robust second order, shock-capturing numerical scheme for multidimensional special relativistic magnetohydrodynamics on computational domains with adaptive mesh refinement is presented. The base solver is a total variation diminishing Lax-Friedrichs scheme in a finite volume setting and is combined with a diffusive approach for controlling magnetic monopole errors. The consistency between the primitive and conservative variables is ensured at all limited reconstructions and the spatial part of the four velocity is used as a primitive variable. Demonstrative relativistic examples are shown to validate the implementation. We recover known exact solutions to relativistic MHD Riemann problems, and simulate the shock-dominated long term evolution of Lorentz factor 7 vortical flows distorting magnetic island chains.     

An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type

This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is not known a priori and the solution is not smooth on it. Accordingly, a very fine grid is needed in order to obtain a solution with a reasonable accuracy. In this paper, our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions. The irregular mesh will be generated such that its density around the free boundary is higher than in other places and high-resolution computations are focused on these zones. To this aim, we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact-supported interpolating wavelets to develop a multi-level algorithm, which generates an adaptive computational grid. Using this adaptive grid takes less CPU time than using a full regular mesh. At each step of the algorithm, the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh. Finally, the numerical examples are presented to show the validity and efficiency of the technique.     

Finite element analysis of triangular fins attached to a thick wall

Heat conduction in an array of triangular fins with an attached wall is modeled using the finite element method. An adaptive mesh refinement technique is developed giving accuracy comparable to uniform mesh refinement and much increased computational efficiency. The effects of wall thickness and fin spacing are examined for various Biot numbers. It is shown that for low Biot numbers (Bi < 0.1), the one-dimensional assumption is valid but for higher Biot numbers (Bi 0.1), two-dimensional heat conduction must be considered, temperature distributions at the fin root are always non-uniform and the fin is found not to be effective.     

A Parallel Particle Tracking Framework for Applications in Scientific Computing

Particle tracking methods are a versatile computational technique central to the simulation of a wide range of scientific applications. In this paper, we present a new parallel particle tracking framework for the applications of scientific computing. The framework includes the in-element particle tracking method, which is based on the assumption that particle trajectories are computed by problem data localized to individual elements, as well as the dynamic partitioning of particle-mesh computational systems. The ultimate goal of this research is to develop a parallel in-element particle tracking framework capable of interfacing with a different order of accuracy of ordinary differential equation (ODE) solver. The parallel efficiency of such particle-mesh systems depends on the partitioning of both the mesh elements and the particles; this distribution can change dramatically because of movement of the particles and adaptive refinement of the mesh. To address this problem we introduce a combined load function that is a function of both the particle and mesh element distributions. We present experimental results that detail the performance of this parallel load balancing approach for a three-dimensional particle-mesh test problem on an unstructured, adaptive mesh, and demonstrate the ability of interfacing with different ODE solvers.     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

A 15-point high-order compact scheme with multigrid computation for solving 3D convection diffusion equations

We propose a method with sixth-order accuracy to solve the three-dimensional (3D) convection diffusion equation. We first use a 15-point fourth-order compact discretization scheme to obtain fourth-order solutions on both fine and coarse grids using the multigrid method. Then an iterative mesh refinement technique combined with Richardson extrapolation is used to approximate the sixth-order accurate solution on the fine grid. Numerical results are presented for a variety of test cases to demonstrate the efficiency and accuracy of the proposed method, compared with the standard fourth-order compact scheme.