On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.     

Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems

In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids), we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.     

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.     

Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

Adaptive moving mesh upwind scheme for the two-species chemotaxis model

Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacteria/cells in response to an external stimulus, usually a chemical one. A common property of all chemotaxis systems is their ability to model a concentration phenomenon—rapid growth of the cell density in small neighborhoods of concentration points/curves. More precisely, the solution may develop singular, spiky structures, or even blow up in finite time. Therefore, the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task.We study the two-species Patlak–Keller–Segel type chemotaxis system, in which the two species do not compete, but have different chemotactic sensitivities, which may lead to a significantly difference in cell density growth rates. This phenomenon was numerically investigated in Kurganov and Luká?ová-Medvi?ová (2014) and Chertock et al. (2018), where second- and higher-order methods on uniform Cartesian grids were developed. However, in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity, a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected.In this work, we consider an alternative approach relying on mesh adaptation, which helps to improve the approximation of the singular structures evolved by chemotaxis models. We develop, in particular, an adaptive moving mesh (AMM) finite-volume semi-discrete upwind method for the two-species chemotaxis system. The proposed AMM technique allows one to increase the density of mesh nodes at the blowup regions. This helps to substantially improve the resolution while using a relatively small number of finite-volume cells.     

On the application of a least squares residual-fitting finite element formulation to fluid flow problems

The connection between the least squares residual fitting finite element formulation and reduced integration is considered for two model problems, one of which requires an isoparametric mapping. For both problems a significant improvement in accuracy is achieved when compared with a conventional Galerkin finite element formulation. It is found that the two methods are equivalent for rectangular elements as long as an isoparametric mapping is not required. For triangular elements there is no direct link between the two methods and neither method produces any significant improvement over the use of exact numerical integration.     

On the employ of meshless methods in biomechanics

In this paper we analyse the convenience and possible advantages of using meshless methods in numerical simulations in biomechanics. While finite elements have been the universal tool during the last decades to perform such simulations, a recently developed wide family of methods, globally coined as meshless methods, has emerged as an attractive choice for an increasing variety of problems. They present some key advantages such as the absence of a mesh in the traditional sense, particularly important in domains of very complex geometry. These methods are also able to easily handle finite strains and large displacements in a Lagrangian framework due to their relatively less sensitivity to the point distributions. In this work we discuss the use of one of them, the natural element method, in biomechanics. After a brief review of this new approach, such a choice is justified and some examples showing the performance of the method are presented.     

Capturing expert knowledge of mesh refinement in numerical methods of impact analysis by means of genetic programming

The mesh refinement decisions of an experienced user of high-velocity impact numerical approximation finite differences computations are discovered as a set of comprehensible rules by means of Genetic Programming. These rules that could automatically trigger adaptive mesh refinement to mimic the expert user, detect mesh cells that require refinement by evolving a formula involving cell quantities such as material densities. Various cell variable combinations are investigated in order to identify the optimal ones for indicating mesh refinement. A high-velocity impact phenomena example of a tungsten ball that strikes a steel plate illustrates this methodology.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

Adaptive finite element mesh triangulation using self-organizing neural networks

The finite element method is a computationally intensive method. Effective use of the method requires setting up the computational framework in an appropriate manner, which typically requires expertise. The computational cost of generating the mesh may be much lower, comparable, or in some cases higher than the cost associated with the numeric solver of the partial differential equations, depending on the application and the specific numeric scheme at hand.The aim of this paper is to present a mesh generation approach using the application of self-organizing artificial neural networks through adaptive finite element computations. The problem domain is initially constructed using the self-organizing neural networks. This domain is used as the background mesh which forms the input for finite element analysis and from which adaptive parameters are calculated through adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust the location of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighborhood of the chosen node. The adjustment is a movement of the selected nodes toward a specific input point on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement of nodal points adjusts the element sizes in a way that the concentration of elements will occur in the regions of high stresses. The methods and experiments developed here are for two-dimensional triangular elements but seem naturally extendible to quadrilateral elements.     

Shape reconstruction on a varying mesh

A central class of image understanding problems is concerned with reconstructing a shape from an incomplete data set, such as fitting a surface to (partially) given contours. A new theory for solving such problems is presented. Unlike the current heuristic methods, the method used starts from fundamental principles that should be followed by any reconstruction method, regardless of its mathematical or physical implementation. A mathematical procedure which conforms to these principles is presented. One major advantage of the method is the ability to handle shapes containing both smooth and sharp parts without using thresholds. A sharp variation, such as a corner, requires a high-resolution mesh for adequate representation, while slowly varying sections can be represented with sparser mesh points. Unlike current methods, this procedure fits the surface on a varying mesh. The mesh is constructed automatically to be more dense at parts of the image that have more rapid variation. Analytical examples are given in simple cases, followed by numerical experiments     

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.     

基于径向基函数与B样条的散乱数据拟合方法

针对散乱数据的曲面拟合问题,提出一种径向基函数与B样条插值结合使用的曲面拟合方法.通过分片径向基函数插值,三维散乱点,再从分片插值曲面上获取预先设定好的有序网格点的值,最后利用张量积B样条插值有序网格点,从而得到拟合曲面.该方法较好地解决散乱数据插值和拟合的计算不稳定性问题,最后给出算法实例.     

Direct Reconstruction of a Displaced Subdivision Surface from Unorganized Points

In this paper we describe the generation of a displaced subdivision surface directly from a set of unorganized points. The displaced subdivision surface is an efficient mesh representation that defines a detailed mesh with a displacement map over a smooth domain surface and has many benefits including compression, rendering, and animation, which overcome limitations of an irregular mesh produced by an ordinary mesh reconstruction scheme. Unlike previous displaced subdivision surface reconstruction methods, our method does not rely on a highly detailed reconstructed mesh. Instead, we efficiently create a coarse base mesh, which is used to sample displacements directly from unorganized points, and this results in a simple process and fast calculation. We suggest a shrink-wrapping-like shape approximation and a point-based mesh simplification method that uses the distance between a set of points and a mesh as an error metric to generate a domain surface that optimally approximates the given points. We avoid time-consuming energy minimization by employing a local subdivision surface fitting scheme. Finally, we show several reconstruction results that demonstrate the usability of our algorithm.     

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints

In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.     

The Superconvergent Cluster Recovery Method

A new gradient recovery technique SCR (Superconvergent Cluster Recovery) is proposed and analyzed for finite element methods. A linear polynomial approximation is obtained by a least-squares fitting to the finite element solution at certain sample points, which in turn gives the recovered gradient at recovering points. Compared with similar techniques such as SPR and PPR, our approach is cheaper and efficient, while having same or even better accuracy. In additional, it can be used as an a posteriori error estimator, which is relatively simple to implement, cheap in terms of storage and computational cost for adaptive algorithms. We present some numerical examples illustrating the effectiveness of our recovery procedure.     

Overview and applications of the reproducing Kernel Particle methods

Summary Multiple-scale Kernel Particle methods are proposed as an alternative and/or enhancement to commonly used numerical methods such as finite element methods. The elimination of a mesh, combined with the properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and high modal density. The Reproducing Kernel Particle Method (RKPM) utilizes the fundamental notions of the convolution theorem, multiresolution analysis and window functions. The construction of a correction function to scaling functions, wavelets and Smooth Particle Hydrodynamics (SPH) is proposed. Completeness conditions, reproducing conditions and interpolant estimates are also derived. The current application areas of RKPM include structural acoustics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity. The effectiveness of RKPM is extended through a new particle integration method. The Kronecker delta properties of finite element shape functions are incorporated into RKPM to develop a C ^m kernel particle finite element method. Multiresolution and hp-like adaptivity are illustrated via examples.     

Accurate finite element modeling of acoustic waves

In the paper we suggest an accurate finite element approach for the modeling of acoustic waves under a suddenly applied load. We consider the standard linear elements and the linear elements with reduced dispersion for the space discretization as well as the explicit central-difference method for time integration. The analytical study of the numerical dispersion shows that the most accurate results can be obtained with the time increments close to the stability limit. However, even in this case and the use of the linear elements with reduced dispersion, mesh refinement leads to divergent numerical results for acoustic waves under a suddenly applied load. This is explained by large spurious high-frequency oscillations. For the quantification and the suppression of spurious oscillations, we have modified and applied a two-stage time-integration technique that includes the stage of basic computations and the filtering stage. This technique allows accurate convergent results at mesh refinement as well as significantly reduces the numerical anisotropy of solutions. We should mention that the approach suggested is very general and can be equally applied to any loading as well as for any space-discretization technique and any explicit or implicit time-integration method.