Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in [25]. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.     

A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes

In this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.     

An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi-implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.     

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

A genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent, i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.     

Runge-Kutta Discontinuous Galerkin Method Using WENO-Type Limiters: Three-Dimensional Unstructured Meshes

This paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.     

A Priori Subcell Limiting Approach for the FR/CPR Method on Unstructured Meshes

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) methods on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based onmodal energy coefficients is proposed to detect troubled cells, where discontinuitiesexist. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing schemebased on nonuniform nonlinear weighted (NNW) interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPRmethod. Numerical investigations show that the proposed subcell limiting strategy onunstructured quadrilateral meshes is robust in shock-capturing.     

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper, we propose a high-order accurate discontinuous Galerkin(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state andprovably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitationon rectangular meshes, the extension to triangular meshes is conceptually plausiblebut highly nontrivial. We first introduce a special way to recover the equilibrium stateand then design a group of novel variables at the interface of two adjacent cells, whichplays an important role in the well-balanced and positivity-preserving properties. Onemain challenge is that the well-balanced schemes may not have the weak positivityproperty. In order to achieve the well-balanced and positivity-preserving propertiessimultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivityproperty. A simple existing limiter can be applied to enforce the positivity-preservingproperty, without losing high-order accuracy and conservation. Extensive one- andtwo-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.     

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG)method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classicalpolynomials of degree $N$ inside each element, in our new approach the discrete solutionis represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DGmethod on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on eachsub-triangle are defined, as usual, on a universal reference element, hence allowing tocompute universal mass, flux and stiffness matrices for the subgrid triangles once andfor all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructurednature of the computational grid. High order of accuracy in time is achieved thanksto the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical resultshave been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtainedby the corresponding modal DG version of the scheme.     

A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems

We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost. The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty. Combined with high accuracy of the gPC-based forward solver, the new algorithm can provide great efficiency in practical applications. A rigorous error analysis of the algorithm is conducted, where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate. It is proved that fast (exponential) convergence of the gPC forward solution yields similarly fast (exponential) convergence of the posterior. The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension.     

A Numerical Comparison Between Quasi-Monte Carlo and Sparse Grid Stochastic Collocation Methods

Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations. These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse grid methods in the context of ground-water flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance.     

A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years. Similar to Monte Carlo methods, the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it. The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method, Smolyak method, Stroud 2 or 3 cubature method, and adaptive Stroud method. Another type of collocation method, the probabilistic collocation method (PCM), has also been proposed and applied to flow in porous media. In this paper, we discuss these methods in terms of their accuracy, efficiency, and applicable range for flow in spatially correlated random fields. These methods are compared in details under different conditions of spatial variability and correlation length. This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency. The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods. Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.     

Meshes in der Bauchwand

Zusammenfassung. Angesichts einer deutlichen Senkung der Rezidivrate erf?hrt die Verst?rkung der Bauchwand mit alloplastischen Materialien, den sogenannten Meshes, eine zunehmende Verbreitung in der Therapie der Narbenhernien. Da die Wiederholung des prim?r versagenden Verfahrens bei über der H?lfte der Patienten zum Rezidivbruch führt, ist die rationale Grundlage der chirurgischen Mesh-Therapie die Notwendigkeit eines Verfahrenswechsels, insbesondere unter Berücksichtigung eines m?glichen zugrundeliegenden Defekts im Kollagenmetabolismus. Nach über 100 Jahren der Mesh-Entwicklung werden diese vorwiegend in Sublay- oder Onlay-Position plaziert, zum Einsatz kommen überwiegend ePTFE, Polyester oder Polypropylen. Stets mu? wegen der obligaten Wundkontraktion eine ausreichende überlappung angestrebt werden. Auf der Grundlage eigener Erfahrungen und der Literatur werden die jeweiligen Vor- und Nachteile der verschiedenen Mesh-Techniken und -Materialien diskutiert. Dabei ist das kumulative Risiko für die Entstehung von Langzeitkomplikationen, wie Mesh-Wanderung, Fistelbildung, das Ausma? Mesh-bedingter Beschwerden oder das potentielle Risiko einer persistierenden Fremdk?rperentzündung wegen fehlender Langzeitstudien bislang nicht absch?tzbar. Insgesamt stellen jedoch angesichts fehlender therapeutischer Alternativen die Meshes einen nicht zu untersch?tzenden Fortschritt in der Hernienchirurgie dar.      

Meshes in der Bauchwand

Angesichts einer deutlichen Senkung der Rezidivrate erf?hrt die Verst?rkung der Bauchwand mit alloplastischen Materialien, den sogenannten Meshes, eine zunehmende Verbreitung in der Therapie der Narbenhernien. Da die Wiederholung des prim?r versagenden Verfahrens bei über der H?lfte der Patienten zum Rezidivbruch führt, ist die rationale Grundlage der chirurgischen Mesh-Therapie die Notwendigkeit eines Verfahrenswechsels, insbesondere unter Berücksichtigung eines m?glichen zugrundeliegenden Defekts im Kollagenmetabolismus. Nach über 100 Jahren der Mesh-Entwicklung werden diese vorwiegend in Sublay- oder Onlay-Position plaziert, zum Einsatz kommen überwiegend ePTFE, Polyester oder Polypropylen. Stets mu? wegen der obligaten Wundkontraktion eine ausreichende überlappung angestrebt werden. Auf der Grundlage eigener Erfahrungen und der Literatur werden die jeweiligen Vor- und Nachteile der verschiedenen Mesh-Techniken und -Materialien diskutiert. Dabei ist das kumulative Risiko für die Entstehung von Langzeitkomplikationen, wie Mesh-Wanderung, Fistelbildung, das Ausma? Mesh-bedingter Beschwerden oder das potentielle Risiko einer persistierenden Fremdk?rperentzündung wegen fehlender Langzeitstudien bislang nicht absch?tzbar. Insgesamt stellen jedoch angesichts fehlender therapeutischer Alternativen die Meshes einen nicht zu untersch?tzenden Fortschritt in der Hernienchirurgie dar.     

Meshes in inguinal hernia repair

Repair of a groin hernia is strongly influenced by prosthetic mesh implantation carried out in nearly 50 % of all operations. Recurrency rates, however, did not decrease by this policy. Many different materials are available. Due to bioinstability on the long-term and elevated infection rates PTFE is not suitable for inguinal hernia repair. Polyester also provides no long-term stability and induces a chronic foreign body reaction. Polypropylene initially leads to an acute inflammatory reaction and often ends in fibrosis. Both reactions are related to the weight of the used mesh. All materials may lead to specific complications. These include seroma formation, infection, migration of the prosthesis with arrosion of organs, damage of the vas deferens, development of recurrency by shrinkage of the fibers around the mesh, formation of adhesions in the preperitoneal position, and chronic inguinal pain. Therefore, meshes should be used only after individual estimation of risks and benefit. This includes the hernia classification, the number of previous operations and the possibility of a defect in collagen metabolism. The unknown long-term risks for the patient may be taken only in strong indications.     

High Order Cubic-Polynomial Interpolation Schemes on Triangular Meshes

The Cubic-Polynomial Interpolation scheme has been developed and applied to many practical simulations. However, it seems the existing Cubic-Polynomial Interpolation scheme are restricted to uniform rectangular meshes. Consequently, this scheme has some limitations to problems in irregular domains. This paper will extend the Cubic-Polynomial Interpolation scheme to triangular meshes by using some spline interpolation techniques. Numerical examples are provided to demonstrate the accuracy of the proposed schemes.     

A Generalised Lattice Boltzmann Equation on Unstructured Grids

This paper presents a new finite-volume discretization of a generalised Lattice Boltzmann equation (LBE) on unstructured grids. This equation is the continuum LBE, with the addition of a second order time derivative term (memory), and is derived from a second-order differential form of the semi-discrete Boltzmann equation in its implicit form. The new scheme, named unstructured lattice Boltzmann equation with memory (ULBEM), can be advanced in time with a larger time-step than the previous unstructured LB formulations, and a theoretical demonstration of the improved stability is provided. Taylor vortex simulations show that the viscosity is the same as with standard ULBE and demonstrates that the new scheme improves both stability and accuracy. Model validation is also demonstrated by simulating backward-facing step flow at low and moderate Reynolds numbers, as well as by comparing the reattachment length of the recirculating eddy behind the step against experimental and numerical data available in literature.     

腹腔镜下回肠袋膀胱造口旁疝补片修补术初步体会

目的 探讨腹腔镜下回肠袋膀胱造口旁疝补片修补术的有效性和安全性.方法 2004年9月至2006年12月,共对18例造口旁疝进行腹腔镜补片修补术,其中回肠袋膀胱造口旁疝3例.结果 3例均修补成功.手术时间45～90 min,平均63.3 min.疝环直径4～5 cm,平均4.5 cm.术后暂时性腹胀1例;1例修补区域腹壁明显疼痛,2周内缓解;1例血清肿,经2次穿刺抽液并加压包扎后治愈,无血肿发生,未发生与手术相关的感染.术后住院时间3～5 d,平均4 d;术后随访17～25个月,平均21.3月,未见复发.结论 腹腔镜下回肠袋膀胱的造口旁疝补片修补术从技术上讲是安全、可行的,早期的临床结果较理想.     

Extended Synchronous Variational Integrators for Wave Propagations on Non-Uniform Meshes

This paper is concerned with the classical problem of wave propagation indiscrete models of nonuniform resolution. We extend the traditional asynchronousvariational integrators (AVIs) method to higher order and couple different spatial elements to adapt to nonuniform meshes. We show that the extension of AVIs methodis stable, convergent and may reduce the spurious inter-grid reflection across mesheswith different sizes. Numerical experiments are provided to verify the stability andconvergence of the extended AVIs. The total energy is numerically conserved in ourexperiments.     

Entropy Stable Scheme on Two-Dimensional Unstructured Grids for Euler Equations

We propose an entropy stable high-resolution finite volume scheme to approximate systems of two-dimensional symmetrizable conservation laws on unstructured grids. In particular we consider Euler equations governing compressible flows. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. High resolution is achieved based on a linear reconstruction procedure satisfying a suitable sign property that helps to maintain entropy stability. The proposed scheme is demonstrated to robustly approximate complex flow features by a series of benchmark numerical experiments.     

Experimental Study of the Effect of Different Meshes on Bacterial Translocation

Implantation of intraabdominal biomaterials promotes bacterial translocation (BT). In this study we planned to determine the effect of different types of mesh on the induction of BT. Swiss albino mice were divided into four groups: control, polypropylene (PP), polyglactin 910 (PG), and dura mater (DM). At hour 0, an abdominal wall defect was created in all animals. In the control group the defect was closed primarily. PP, PG, and DM meshes were used to repair the defects such that the intestines were in contact with the mesh in the remaining three groups. BT was evaluated 4, 24, and 48 hours later. At 4 hours total BT increased in the PP group compared with that in the control (p= 0.0321) and DM (p= 0.0098) groups. At 24 hours the PG group had increased BT compared with the controls (p= 0.0392) and the DM group (p= 0.0274), whereas the PP group had increased BT compared with the DM group only (p= 0.0477). At 48 hours both PG and PP groups had increased BT compared with controls (p= 0.0431 and p= 0.0001, respectively); the PP group had also increased BT compared to the DM group (p= 0.001) and the PG group (p= 0.017). In the control, DM, and PP groups there was no intragroup statistical difference. In the PG group there was a significant increase in BT at 24 hours compared to that at 4 hours (p= 0.0274). Prosthetic meshes led to increased BT compared to that with the DM mesh. This effect might be attributed to the different organic and physical properties of the meshes.