Extremum-Preserving Correction of the Nine-Point Scheme for Diffusion Equation on Distorted Meshes

In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes.We introduce a new nonlinear approach to construct the conservative flux,that is,a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted method.Our new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns,so it can deal with the problem with general discontinuous coefficients.Numerical results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.

Analysis of the nonlinear scheme preserving the maximum principle for the anisotropic diffusion equation on distorted meshes

In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constraints on the cell deformation and the diffusion coefficient.Numerical results show that the scheme is indeed coercive and satisfies the discrete maximum principle,and the accuracy of this scheme is remarkably better than that of an existing scheme preserving the discrete maximum principle on random triangular meshes.

A geometrically and locally adaptive remeshing method for finite difference modeling of mining-induced surface subsidence

Surface subsidence induced by underground mining is a typical serious geohazard.Numerical approaches such as the discrete element method(DEM)and finite difference method(FDM)have been widely used to model and analyze mining-induced surface subsidence.However,the DEM is typically computationally expensive,and is not capable of analyzing large-scale problems,while the mesh distortion may occur in the FDM modeling of largely deformed surface subsidence.To address the above problems,this paper presents a geometrically and locally adaptive remeshing method for the FDM modeling of largely deformed surface subsidence induced by underground mining.The essential ideas behind the proposed method are as follows:(i)Geometrical features of elements(i.e.the mesh quality),rather than the calculation errors,are employed as the indicator for determining whether to conduct the remeshing;and(ii)Distorted meshes with multiple attributes,rather than those with only a single attribute,are locally regenerated.In the proposed method,the distorted meshes are first adaptively determined based on the mesh quality,and then removed from the original mesh model.The tetrahedral mesh in the distorted area is first regenerated,and then the physical field variables of old mesh are transferred to the new mesh.The numerical calculation process recovers when finishing the regeneration and transformation.To verify the effectiveness of the proposed method,the surface deformation of the Yanqianshan iron mine,Liaoning Province,China,is numerically investigated by utilizing the proposed method,and compared with the numerical results of the DEM modeling.Moreover,the proposed method is applied to predicting the surface subsidence in Anjialing No.1 Underground Mine,Shanxi Province,China.

Predicting impact strength of perforated targets using artificial neural networks trained on FEM-generated datasets

The paper considers application of artificial neural networks(ANNs)for fast numerical evaluation of a residual impactor velocity for a family of perforated PMMA(Polymethylmethacrylate)targets.The ANN models were trained using sets of numerical results on impact of PMMA plates obtained via dynamic FEM coupled with incubation time fracture criterion.The developed approach makes it possible to evaluate the impact strength of a particular target configuration without complicated FEM calculations which require considerable computational resources.Moreover,it is shown that the ANN models are able to predict results for the configurations which cannot be processed using the developed FEM routine due to numerical instabilities and errors:the trained neural network uses information from successful computations to obtain results for the problematic cases.A simple static problem of a perforated plate deformation is discussed prior to the impact problem and preferable ANN architectures are presented for both problems.Some insight into the perforation pattern optimization using a genetic algorithm coupled with the ANN is also made and optimized perforation patterns which theoretically enhance the target impact strength are constructed.

A novel twice-interpolation finite element method for solid mechanics problems

Formulation and numerical evaluation of a novel twice-interpolation finite element method （TFEM） is presented for solid mechanics problems. In this method, the trial function for Galerkin weak form is constructed through two stages of consecutive interpolation. The primary interpolation follows exactly the same procedure of standard FEM and is further reproduced according to both nodal values and averaged nodal gradients obtained from primary interpolation. The trial functions thus constructed have continuous nodal gradients and contain higher order polynomial without increasing total freedoms. Several benchmark examples and a real dam problem are used to examine the TFEM in terms of accuracy and convergence. Compared with standard FEM, TFEM can achieve significantly better accuracy and higher convergence rate, and the continuous nodal stress can be obtained without any smoothing operation. It is also found that TFEM is insensitive to the quality of the elemental mesh. In addition, the present TFEM can treat the incompressible material without any modification.

A novel four-node quadrilateral element with continuous nodal stress

Formulation and numerical evaluation of a novel four-node quadrilateral element with continuous nodal stress（Q4-CNS）are presented.Q4-CNS can be regarded as an improved hybrid FE-meshless four-node quadrilateral element（FE-LSPIM QUAD4）, which is a hybrid FE-meshless method.Derivatives of Q4-CNS are continuous at nodes, so the continuous nodal stress can be obtained without any smoothing operation.It is found that,compared with the standard four-node quadrilateral element（QUAD4）,Q4- CNS can achieve significantly better accuracy and higher convergence rate.It is also found that Q4-CNS exhibits high tolerance to mesh distortion.Moreover,since derivatives of Q4-CNS shape functions are continuous at nodes,Q4-CNS is potentially useful for the problem of bending plate and shell models.

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys.285（2015）,265-279] on uniform meshes,in this paper,in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme（UGKS）,we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations.We can prove that the scheme is asymptotic preserving,and especially for the distorted quadrilateral meshes,a nine-point scheme [SIAM J.SCI.COMPUT.30（2008）,1341-1361] for the diffusion limit equations is obtained,which is naturally reduced to standard five-point scheme for the orthogonal meshes.The numerical examples on distorted meshes are included to validate the current approach.

Bubble-Enriched Smoothed Finite Element Methods for Nearly-Incompressible Solids

This work presents a locking-free smoothed finite element method(S-FEM)for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity.The proposed method overcomes well-known issues of standard finite element methods(FEM)in the incompressible limit:the over-estimation of stiffness and sensitivity to severely distorted meshes.The concepts of cell-based,edge-based and node-based S-FEMs are extended in this paper to three-dimensions.Additionally,a cubic bubble function is utilized to improve accuracy and stability.For the bubble function,an additional displacement degree of freedom is added at the centroid of the element.Several numerical studies are performed demonstrating the stability and validity of the proposed approach.The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method.

A PENALTY-EQUILIBRATING HYBRID STRESS 3-D ELEMENT

This paper presents a theory about penalty-equilibrating(PEQ)hybrid element of three dimensions (3-D), and generates a model forthe PEQ hybrid stress 3-D element. By the PEQ ap- proach, the falsestress is avoided and the precision in calculation is raised to alarge extent under the mesh distortion condition without anotheradditional degree of freedom. In the results of numerical ex- amples,the present element is compared with the 8-node hexahedron elementand the optimized hybrid element, and it is proved that ourconclusion is correct. In addition, the penalty-equilibrating hybrid3-D element is taken as a trial to calculate problems of square platebending and incompressibility. The results obtained are satisfactory.

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.

Exploring the efficiency of image metric for assessing the visual quality of 3D mesh model

Recent developments in 3D graphics technology have led to extensive processes on 3D meshes(e.g.,compression,simplification,transmission and watermarking),these processes unavoidably cause the visual perceptual degradation of the 3D objects.The existing mesh visual quality evaluation metrics either require topology constrain or fail to reflect the perceived visual quality.Meanwhile,for the 3D objects that are observed on 2D screens by the users,it is reasonable to apply image metric to assess the distortion caused by mesh simplification.We attempt to explore the efficiency of image metric for assessing the visual fidelity of the simplified 3D model in this paper.For this purpose,several latest and most effective image metrics,2D snapshots,number and pooling algorithms are involved in our study,and finally tested on the IEETA simplification database.The statistical data allow the researcher to select the optimal parameter for this image-based mesh visual quality assessment and provide a new perspective for the design and performance assessment of mesh simplification algorithms.