Mesh Partitioning Research Articles

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.

Tree shrew models: a chronic social defeat model of depression and a one-trial captive conditioning model of learning and memory.

Recent genome studies indicate that tree shrew is in the order or a closest sister of primates, and thus may be one of the best animals to model human diseases. In this paper, we report on a social defeat model of depression in tree shrew (Tupaia belangeri chinensis). Two male tree shrews were housed in a pair-cage consisting of two independent cages separated by a wire mesh partition with a door connecting the two cages. After one week adaptation, the connecting door was opened and a brief fighting occurs between the two male tree shrews and this social conflict session consisted of 1 h direct conflict (fighting) and 23 h indirect influence (e.g. smell, visual cues) per day for 21 days. The defeated tree shrew was considered the subordinate. Compared with naive animals, subordinate tree shrews at the final week of social conflict session showed alterations in body weight, locomotion, avoidance behavior and urinary cortisol levels. Remarkably, these alterations persisted for over two weeks. We also report on a novel captive conditioning model of learning and memory in tree shrew. An automatic trapping cage was placed in a small closed room with a freely-moving tree shrew. For the first four trials, the tree shrew was not trapped when it entered the cage and ate the bait apple, but it was trapped and kept in the cage for 1 h on the fifth trial. Latency was defined as the time between release of the tree shrew and when it entered the captive cage. Latencies during the five trials indicated adaptation. A test trial 24 h later was used to measure whether the one-trial trapping during the fifth trial could form captive memory. Tree shrews showed much longer trapping latencies in the test trial than the adaptation trials. The N-methyl-d-aspartate (NMDA) receptor antagonist MK-801 (0.2 mg/kg, i.p.), known to prevent the formation of memory, did not affect latencies in the adaptation trails, but did block captive memory as it led to much shorter trapping latencies compared to saline treatment in the test trial. These results demonstrate a chronic social defeat model of depression and a novel one-trial captive conditioning model for learning and memory in tree shrews, which are important for mechanism studies of depression, learning, memory, and preclinical evaluation for new antidepressants.

Polygonal Mesh Partitioning for NURBS Surface Generation

Surface reconstruction and NURBS surface generation based on 3D surface mesh partitioning are more essential today. In this paper, we present a new method of automatic partitioning complex surface meshes into the bounded regions with four corner points (quadrilateral regions) based on using control points (notches) for NURBS surface generation. The procedure of this method consists of 4 major steps: (1) the 3D polygons mapping into 2D polygons; (2) convex decomposition of the polygons in the 2D space; (3) subdivision of each polygons into quadrilateral regions; (4) mapping the received 2D regions onto the 3D original surface mesh. Main contribution in this paper is automatic partitioning of the 3D segmented parts of complex surfaces into quadrilateral regions based on combination of segmentation, mapping, and subdivision techniques. Automatic partitioning allows us to create not rectangular but quadrilateral regions without using any user-dependent parameters for further NURBS surface generation.

Non-Equidistant Numerical Methods for Ordinary Differential Equation with Periodical Boundary Value

Ordinary differential equation with periodical boundary value and small parameter multiplied in the highest derivative was considered. The solution of the problem has boundary layers, which is thin region in the neighborhood of the boundary of the domain. Firstly, the properties of boundary layer were discussed. The solution was decomposed into the smooth component and the singular component. The derivatives of the smooth component and the singular component were estimated. Secondly, mesh partition techniques were presented according to one transition point method and multi-transition points method. Thirdly numerical methods based on non-equidistant mesh partition were presented to solve the problem. Finally error estimations were given for both computational methods.

Realization of Quadrilateral Mesh Partition and Optimization Algorithm Based on Cloud Data

Algorithms of quadrilateral mesh partition and optimization based on cloud data are proposed in the paper. Quadrilateral meshes are partite by means of dynamic edge-extending according to the types of the nodes. The mesh optimization algorithms are designed, and topology optimization of degree 2 and 3 node and region mesh optimization processing based on viewpoint are programmed and debugged as well. Realizations of the algorithm are introduced in details. The algorithm improves mesh quality and examples are shown to illustrate its feasibility.

Cooperative behavior of laboratory rats (Rattus norvegicus) in an instrumental task.

Cooperation is a cognitively demanding, complex social behavior, found primarily in primates. Here we investigated mutualism in rats (Rattus Norvegicus), a simple form of cooperation in which two subjects work on operant task, receiving immediate and simultaneous sucrose reward for a joint action. To receive the sucrose reward, familiar pairs of rats were required to nose poke simultaneously. Following 44 training days, we examined the relation of social contact and ultrasonic vocalizations to the rat's cooperative behavior by testing the effects of inserting opaque, wire-mesh, or no partition--between subjects. Cooperative behavior (simultaneous nose-poking): (a) increased gradually during initial training; (b) decreased with the opaque partition (restricting visual, acoustic, and physical communication); (c) increased with a wire mesh partition restricting only physical contact); and (d) increased with the number of 50 kHz USV "happy" calls and the intensity of social interaction. The possibility of studying the development of cooperative behavior in laboratory rats using a simple procedure based on commercially available equipment may prove useful in modeling determinants of social behavior.

Mesh Partitioning Optimization in Parallel Substructure Method Using Improved HYT Approach

This paper proposes an improved HYT (IHYT) approach to achieve better load balance among substructures for better efficiency of the parallel substructure method. In the IHYT method, the matrix ordering method in SPOOLES, which provides the estimated statistics of numerical workload and nonzero entries of the matrix, are employed for predicting the workload of each sub-mesh, and used as the basis for tuning the element weights to embed the workload prediction phase. In addition, a partitioning kernel–JOSTLE, which can slightly tune the partitioning result from the last iteration and is good for the convergence of iterations, are employed as the partitioning kernel. The numerical experiments show that the IHYT further achieves better load balance of substructure condensation and leads to better efficiency of the parallel substructure method than HYT approach.

Is full physical contact necessary for buffering effects of pair housing on social stress in rats?

Our previous study showed that pair housing with a familiar male prevented an increase in anxiety caused by social defeat in male rats. The present study attempted to identify the aspects of social interactions with a familiar male that are needed for the emergence of such a pair-housing effect. In Experiment 1, the subject was repeatedly exposed to the cage and bedding used by a familiar pairmate, after two instances of social defeat. Mere exposure to the soiled cage and bedding did not prevent an increase in anxiety in the elevated plus-maze test performed two weeks after social defeat. In Experiment 2, the subject was separated from a familiar pairmate with a wire mesh partition, which allowed visual, auditory, and limited physical contact, in addition to olfactory contact with the pairmate. The separation with a wire mesh partition abolished the buffering effect of pair housing on anxiety. These results indicate that visual, auditory, and olfactory contact with a familiar male was not sufficient in reducing the anxiogenic effect of social defeat in male rats. It was suggested that full physical contact is necessary for the emergence of the buffering effect of pair housing on social stress.

Unstructured mesh partition improvement for implicit finite element at extreme scale

Parallel simulations at extreme scale require that the mesh is distributed across a large number of processors with equal work load and minimum inter-part communications. A number of algorithms hav...

A parallel polynomial Jacobi–Davidson approach for dissipative acoustic eigenvalue problems

We consider a rational algebraic large sparse eigenvalue problem arising in the discretization of the finite element method for the dissipative acoustic model in the pressure formulation. The presence of nonlinearity due to the frequency-dependent impedance poses a challenge in developing an efficient numerical algorithm for solving such eigenvalue problems. In this article, we reformulate the rational eigenvalue problem as a cubic eigenvalue problem and then solve the resulting cubic eigenvalue problem by a parallel restricted additive Schwarz preconditioned Jacobi–Davidson algorithm (ASPJD). To validate the ASPJD-based eigensolver, we numerically demonstrate the optimal convergence rate of our discretization scheme and show that ASPJD converges successfully to all target eigenvalues. The extraneous root introduced by the problem reformulation does not cause any observed side effect that produces an undesirable oscillatory convergence behavior. By performing intensive numerical experiments, we identify an efficient correction-equation solver, an effective algorithmic parameter setting, and an optimal mesh partitioning. Furthermore, the numerical results suggest that the ASPJD-based eigensolver with an optimal mesh partitioning results in superlinear scalability on a distributed and parallel computing cluster scaling up to 192 processors.

Compression of 3D Mesh Geometry and Vertex Attributes for Mobile Graphics

This paper presents a compression scheme for mesh geometry, which is suitable for mobile graphics. The main focus is to enable real-time decoding of compressed vertex positions while providing reasonable compression ratios. Our scheme is based on local quantization of vertex positions with mesh partitioning. To prevent visual seams along the partitioning boundaries, we constrain the locally quantized cells of all mesh partitions to have the same size and aligned local axes. We propose a mesh partitioning algorithm to minimize the size of locally quantized cells, which relates to the distortion of a restored mesh. Vertex coordinates are stored in main memory and transmitted to graphics hardware for rendering in the quantized form, saving memory space and system bus bandwidth. Decoding operation is combined with model geometry transformation, and the only overhead to restore vertex positions is one matrix multiplication for each mesh partition. In our experiments, a 32-bit floating point vertex coordinate is quantized into an 8-bit integer, which is the smallest data size supported in a mobile graphics library. With this setting, the distortions of the restored meshes are comparable to 11-bit global quantization of vertex coordinates. We also apply the proposed approach to compression of vertex attributes, such as vertex normals and texture coordinates, and show that gains similar to vertex geometry can be obtained through local quantization with mesh partitioning.

Establishment and validation of 3-dimensional finite element model of Lenke 1 BN idiopathic scoliosis

Objective To set up and validate three-dimensional finite element model (FEM) of Lenke 1 BN idiopathic scoliosis based on CT images, for building ideal digitization platform for further biomechanical study. Methods An 18-year-old female Lenke 1 BN idiopathic scoliosis patient was selected as volunteer for current study. CT transverse scanning in supine position was done from T1 to caudal end in 1mm layer interval. All CT images were imported into Mimics 10.01 to form qualified AIS three-dimensional geometric model after geometry clean, including all thoraco-lumbar-sacral vertebrae and thoracic cage, which was further delivered to Hypher Mesh 7.0 to build 3d finite element AIS model by mesh partition and quality control. A variety of material parameters were given to different mesh according to references. Validation verification was done by left and right Bending test and erect-supine test. Results 1 )A complete three-dimensional finite element model of Lenke 1 BN idiopathic scoliosis was built successfully, in consist of 341 228 nodes,1 409 929 tetrahedron elements, 163 132 shell elements, 715 cable elements and 149 rod elements. 2)The convex bending Cobb's angles of PT, MT and L curve on X-ray films and finite element simulation were 14°,26° ,8° and 15° ,24°,6° respectively. The error of finite element simulation was 8.3%. 3)Erect Cobb's angles of PT, MT and L were 37°, 50°, 24° and 33°, 51°, 24° on X-ray films and by finite element simulation.Supine Cobb's angles of PT, MT and L were 29°, 43°, 22° and 27°, 42°, 22° on X-ray films and by finite element simulation. The average error of finite element simulation was 3.9%. Conclusion Lenke 1 BN idiopathic scoliosis FEM is intact, reliable and effective for further biomechanical simulation study. Key words: Scoliosis; Finite element analysis; Biomechanics

Asymptotic Analysis for Hyperbolic Equation with Neumann Condition

The second-order hyperbolic equation with small parameter and Neumann con- dition is considered. This kind of problem loses the boundary conditions both in x = 0 and x = 1, while it also loses two initial boundary conditions in t = 0. The solution changes rapidly near two boundary layers and one initial layer. Firstly, the asymptotic solution was studied. The analytical solution was approximated by the degenerate solution and two boundary layer functions and one initial layer function. Secondly, three transition points were presented ac- cording to Shishkin’s idea. Non-equidistant mesh partitions both in x direction and t direction were introduced. An effective computational method is given according to non-equidistant mesh partitions. Finally, numerical experiment was given.

A 2D block-structured mesh partitioner for accurate flow simulations on non-rectangular geometries

The motivation of this work is to carry out parallel simulations of incompressible flows on block-structured meshes. A new partitioning method is proposed. The quality of rectangular partitions is checked and compared with other methods, as regards load balance, edge-cut and block numbers. The partitioner is coupled with the massively parallel Hypre solver library and efficiency of the coupling is measured. Finally, the code is applied to study laminar flows (steady and unsteady) on three non-rectangular geometries. Very fine grids are used to compute reference solutions of a Z-shaped channel flow and the L-shaped and double lid driven cavities.

Note: An original system to separately control litter and female feed intake without modification of the mother-young relations.

This paper describes an original system allowing separate control of the feed intake of the mother and of her litter without rearing them in separate cages, and thus without modifying young / mother relations. This system involved several modifications on tradilional wire cages. Firstly, the female's feeder was modified to prevent the ingestion of pellets by the suckling rabbits. Secondly, young had access to a specific feeder. Lastly, a wire mesh partition divided the cage in two areas, to avoid the doe having access to the young's feeder. Females need to be adapted to their new feeder at least two weeks befare the beginning of the control of feed intake. About 10% of the females did not adapt to this system while the others recovered their initial feed intake within one week. Neither the milk intake nor the growth of the suckling rabbits were modified by this system. Therefore, it allows independent feeding plans for females and litters.

High-performance Computing to Simulate Large-scale Industrial Flows in Multistage Compressors

The aim of this study is to propose a computing method to obtain a detailed simulation of the unsteady flow that develops in multistage turbomachines. The three-dimensional unsteady Reynolds-averaged Navier—Stokes equations are solved using a structured multiblock decomposition method. Although this kind of flow solver is very popular in the turbomachine community nowadays, the complex block connectivities used in meshes of industrial configurations can be penalizing for parallel computing. The computing strategy implemented in a modern flow solver is investigated in this paper, with a particular interest in mesh partitioning, communications and load balancing. Advantages and drawbacks of different computing platforms are then discussed, ranging from vector supercomputers to massively scalar platforms. Comparisons are performed regarding criteria such as the elapsed time and the electrical power consumption. The results show that the use of a large number of computing cores (>128) is heavily penalized by communications and load balancing errors, whereas computing performance with a moderate number of computing cores (<128) is mainly driven by the peak power of the architecture. To help users estimate a priori the parallel performance of a task, a tool based on an extension of Amdahl’s law is proposed, showing satisfying results when compared with observations. Finally, an unsteady flow simulation is performed in a complete three-stage compressor at the design operating point. While still far beyond industrial resources, this numerical flow simulation shows that potential breakthroughs in the design of compression systems can be expected.

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

The parallelization of a fully implicit and stable finite element algorithm with relative low memory requirements for the accurate simulation of time-dependent, free-surface flows of multimode viscoelastic liquids is presented. It is an extension of our multi-stage sequential solution procedure which is based on the mixed finite element method for the velocity and pressure fields, an elliptic grid generator for the deformation of the mesh, and the discontinuous Galerkin method for the viscoelastic stresses [Dimakopoulos and Tsamopoulos [12,14]]. Each one of the above subproblems is solved with the Newton–Rapshon technique according to its particular characteristics, while their coupling is achieved through Picard cycles. The physical domain is graphically partitioned into overlapping subdomains. In the process, two different kinds of parallel solvers are used for the solution of the distributed set of flow and mesh equations: a multifrontal, massively parallel direct one ( MUMPS) and a hierarchical iterative parallel one ( HIPS), while viscoelastic stress components are independently calculated within each finite element. The parallel algorithm retains all the advantages of its sequential predecessor, related with the robustness and the numerical stability for a wide range of levels of viscoelasticity. Moreover, irrespective of the deformation of the physical domain, the mesh partitioning remains invariant throughout the simulation. The solution of the constitutive equations, which constitutes the largest portion of the system of the governing, non-linear equations, is performed in a way that does not need any data exchange among the cluster's nodes. Finally, indicative results from the simulation of an extensionally thinning polymeric solution, demonstrating the efficiency of the algorithm are presented.

Controlling Unstructured Mesh Partitions for Massively Parallel Simulations

Parallel simulations at extreme scale require that the mesh is distributed across a large number of processors with equal work load and minimum interpart communications. A number of algorithms have been developed to meet these goals, e.g., graph/hypergraph and coordinate-based methods. However, the global implementation of current approaches can fail on very large core counts, which is resolved by combining global and local partitioning using multiple parts per processor. The other limitation of graph/hypergraph-based partitioning is that it uses one type of mesh entity as graph nodes; thus, the balance of other mesh entities may not be optimal. In the case of three-dimensional (3-D) linear finite element analysis, it is common to select mesh regions (elements) as partition objects. In current examples, the regions are well balanced up to 163,840 parts for a 1.07 billion element mesh, while the vertices have an imbalance which is as high as 19.52%. Two methods are developed that work in conjunction with graph/hypergraph-based procedures to provide improved partitions. Example computations executed on an IBM Blue Gene/P system using up to 163,840 cores demonstrate the usefulness of the procedures, particularly for time-critical calculations where individual cores may be lightly loaded in terms of the number of mesh entities per core. The algorithms presented in this paper reduced the vertex imbalance from 17.8% to 4.97% for a partition with 131,072 parts and accelerated the equation solution phase of the finite element analysis by 10.4%.

Space–time adaptive numerical methods for geophysical applications

In this paper we present high-order formulations of the finite volume and discontinuous Galerkin finite-element methods for wave propagation problems with a space-time adaptation technique using unstructured meshes in order to reduce computational cost without reducing accuracy. Both methods can be derived in a similar mathematical framework and are identical in their first-order version. In their extension to higher order accuracy in space and time, both methods use spatial polynomials of higher degree inside each element, a high-order solution of the generalized Riemann problem and a high-order time integration method based on the Taylor series expansion. The static adaptation strategy uses locally refined high-resolution meshes in areas with low wave speeds to improve the approximation quality. Furthermore, the time step length is chosen locally adaptive such that the solution is evolved explicitly in time by an optimal time step determined by a local stability criterion. After validating the numerical approach, both schemes are applied to geophysical wave propagation problems such as tsunami waves and seismic waves comparing the new approach with the classical global time-stepping technique. The problem of mesh partitioning for large-scale applications on multi-processor architectures is discussed and a new mesh partition approach is proposed and tested to further reduce computational cost.

Advanced parallel computing in material forming with CIMLib

This paper presents a fully parallel multi-component Library called CIMLib. CIMLib contains a set of components that allow to build efficiently numerical simulation of a various processes mainly in material forming. We describe in this paper the main components of the library: parallel mesh partitioning, parallel remeshing, the Finite Element modeling and the parallel storage and visualization. Two large numerical simulations are presented: the first one focuses on a multi-bodies contact problem, including friction, for complex 3D forming processes. The mesh is evolving during the simulation from 52K nodes to 7M nodes and 64 cores are used to handle this application. The second simulation concerns the multiphase problems involved in the manufacturing processes of full parts. The simulation is done using 88 processors and the mesh is refined during the simulation the final mesh has over 25M nodes.