3D Benchmark Research Articles

Implementation of the CPU/GPU hybrid parallel method of characteristics neutron transport calculation using the heterogeneous cluster with dynamic workload assignment

In recent years, graphics processing units (GPUs) have been adopted in many High-Performance Computing (HPC) systems due to their massive computational power and superior energy efficiency. And accelerating CPU-version computational code on heterogeneous clusters with multi-core CPUs and GPUs has attracted a lot of attention. One of the focus on heterogeneous computing is to efficiently take advantage of all computational resources, including both CPU and GPU available on a cluster. In this paper, a heterogeneous MPI + OpenMP/CUDA parallel algorithm for solving the 2D neutron transport equation with the method of characteristic (MOC) is implemented. In this algorithm, the spatial domain decomposition technique provides the coarse-grained parallelism with the MPI protocol while the fine-grained parallelism is exploited through OpenMP (in CPU calculated domain) and CUDA (in GPU calculated domain) based on the ray parallelization. In order to efficiently leverage the computing power of heterogeneous clusters, a dynamic workload assignment scheme is proposed, which is to distribute the workload based on the runtime performance of CPUs and GPUs in the cluster. Moreover, the strong scaling performance of the MPI + CUDA parallelization is studied through a performance analysis model which provides the detailed impact of the degradation in iteration scheme, the load imbalance issue, the data copy between CPUs and GPUs, and the MPI communication in the MPI + CUDA parallel algorithm. And the corresponding conclusion is still tenable for the MPI + OpenMP/CUDA parallelization. The C5G7 2D benchmark and an extended 2D whole-core problem are calculated with MPI + CUDA parallelization, MPI + OpenMP/CUDA parallelization, and the MPI parallelization for comparison. Numerical results demonstrate that the heterogeneous parallel algorithm maintains the desired accuracy. And the dynamic workload assignment scheme can provide the optimal workload assignment which ideally matches the experimental results. In addition, over 11% improvement is observed in MPI + OpenMP/CUDA parallelization compared against the MPI + CUDA parallelization. Moreover, the CPUs/GPUs heterogeneous clusters significantly outperform the CPUs clusters and one heterogeneous node shows basically five times faster than a CPUs node.

Efficient and conservative fluids using bidirectional mapping

In this paper, we introduce BiMocq 2 , an unconditionally stable, pure Eulerianbased advection scheme to efficiently preserve the advection accuracy of all physical quantities for long-term fluid simulations. Our approach is built upon the method of characteristic mapping (MCM). Instead of the costly evaluation of the temporal characteristic integral, we evolve the mapping function itself by solving an advection equation for the mappings. Dual mesh characteristics (DMC) method is adopted to more accurately update the mapping. Furthermore, to avoid visual artifacts like instant blur and temporal inconsistency introduced by re-initialization, we introduce multi-level mapping and back and forth error compensation. We conduct comprehensive 2D and 3D benchmark experiments to compare against alternative advection schemes. In particular, for the vortical flow and level set experiments, our method outperforms almost all state-of-art hybrid schemes, including FLIP, PolyPic and Particle-Level-Set, at the cost of only two Semi-Lagrangian advections. Additionally, our method does not rely on the particle-grid transfer operations, leading to a highly parallelizable pipeline. As a result, more than 45× performance acceleration can be achieved via even a straightforward porting of the code from CPU to GPU.

A variable horizon method for coupling meshfree peridynamics to FEM

Peridynamics is a nonlocal reformulation of solid mechanics which can reproduce the solution of classical continuum mechanics as its horizon approaches zero. Taking this idea into consideration, we proposed a new coupling approach between meshfree peridynamics and finite element method. An intermediate variable horizon element called “morakkab” is introduced to transfer force and momentum between peridynamic and FEM regions. In the original peridynamic formulation, artificial ghost forces will arise if the horizon varies with position; The creation of these artifacts is avoided by utilizing a modified peridynamic equation of motion based on the concept of co-family. The accuracy of the method is illustrated in 2D benchmark tests. Using the modified equation of motion, compared to the original peridynamic formulation, dramatically decreases the relative errors with respect to the analytical solution.

Fifth order finite volume WENO in general orthogonally - curvilinear coordinates

A fifth order finite volume WENO reconstruction scheme is proposed in the framework of orthogonally - curvilinear coordinates for solving hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values (Q¯i) to reconstruct the right (qi+), middle (qiM), and left (qi−) interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermonde - like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly - /irregularly - spaced grid in orthogonally - curvilinear coordinates is proposed. A grid independent relation for evaluating the smoothness indicator is derived from the basic definition. Finally, a computationally efficient extension to multi - dimensions is proposed along with the procedures for flux and source term integrations. Analytical values of the linear weights, optimal weights, and weights for flux and source term integrations are provided for a regularly - spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO - JS can be fully recovered in the case of limiting curvature (R → ∞). The fifth order finite volume WENO - C (orthogonally - curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark tests involving smooth and discontinuous flows in cylindrical and spherical coordinates.

A time-space flux-corrected transport finite element formulation for solving multi-dimensional advection-diffusion-reaction equations

We present a time-space flux-corrected transport (FCT) finite element formulation for the multi-dimensional time-dependent advection-diffusion-reaction equation. Monotonic solutions can be achieved with the presented method while large time steps (with Courant number Cr>1) are used. Numerical verification, as well as a grid convergence analysis, are carried out for 1D and 2D benchmark problems. A 1D Burgers' equation with various Reynolds numbers is solved with the time-space FCT method. In addition, a 2D transport problem with (and without) a nonlinear reaction inside of a cavity is considered. Finally, the newly developed time-space FCT formulation is applied for modeling biofilm growth problems based on a continuum mathematical model. It turns out that the time-space FCT method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time-space finite incremental calculus (FIC) method.

3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers

This work presents a modeling approach for single-phase flow in 3D fractured porous media with non-conforming meshes. To this end, a Lagrange multiplier method is combined with a parallel variational transfer approach. This Lagrange multiplier method enables the use of non-conforming meshes and depicts the variable coupling between fracture and matrix domain. The variational transfer allows general, accurate, and parallel projection of variables between non-conforming meshes (i.e. between fracture and matrix domain). Comparisons of simulations with 2D benchmarks show good agreement, and the applied finite element Lagrange multiplier spaces show good performance. The method is further evaluated on 3D fracture networks by comparing it to results from conforming mesh simulations which were used as a reference. Application to realistic fracture networks with hundreds of fractures is demonstrated. Mesh size and mesh convergence are investigated for benchmark cases and 3D fracture network applications. Results demonstrate that the Lagrange multiplier method, in combination with the variational transfer approach, is capable of modeling single-phase flow through realistic 3D fracture networks.

A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

SummaryThis work introduces an innovative parallel fully‐distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well known that virtual part design and qualification in additive manufacturing requires highly accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time‐to‐market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high‐end distributed‐memory machines. It is grounded on three building blocks: (1) hierarchical adaptive mesh refinement with octree‐based meshes; (2) a parallel strategy to model the growth of the geometry; and (3) state‐of‐the‐art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder‐bed technologies. After verification against a three‐dimensional (3D) benchmark, a strong‐scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part‐scale simulations of metal or polymer additive manufacturing but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time.

Transport consistent diffusion coefficient for CMFD acceleration and comparison of convergence properties

ABSTRACT A diffusion coefficient for the coarse mesh finite difference (CMFD) acceleration is derived from the semi-analytic solution of one-group, one-dimensional, even-parity transport equation. The derived diffusion coefficient, i.e., the transport consistent diffusion coefficient (TCD), depends on the optical length of a mesh and shows similar behavior with the artificial grid diffusion (AGD) and the effective diffusion (EffD) coefficients for an optically thick mesh. Convergence properties of typical diffusion coefficients are evaluated using the linearized Fourier analysis. Analyses of the C5G7 3D benchmark problems with and without voided region are carried out to compare the convergence properties. The number of transport sweeps to reach convergence using TCD is smaller than that using EffD or AGD.

Accurate discretization of diffusion in the LS-STAG cut-cell method using diamond cell techniques

In cut-cell methods for fluid flow in complex geometries such as the LS-STAG method (Nikfarjam et al. Comput Phys Commun 226:67-80, 2018), mesh non-orthogonality near the immersed boundary makes inaccurate 2-point formulas for computing the face-normal gradients in diffusive fluxes. A way to improve the accuracy is to compute the whole solution gradient at the cut-cell faces, thus decomposing the flux as an orthogonal contribution (using a standard 2-point formula) and non-orthogonal correction (using data at cell vertices). The solution at cut-cell vertices is then interpolated from cell-centered data and boundary conditions. This gradient reconstruction technique is commonly denominated “diamond cell method”.The diamond cell method is implemented in the LS-STAG code for cell-centered data (temperature equation in 2D geometries) and face-centered data (Navier-Stokes equations in 3D-extruded geometries) using various interpolation schemes for vertex reconstruction. The accuracy of the diamond cell discretization is firmly assessed on a series of 2D benchmark problems (Taylor-Couette solution, natural convection from a cylinder in an enclosure) by inspecting the formal order of accuracy and the heat flux distribution at the immersed boundary. Finally, the diamond cell technique is employed for enhancing the accuracy of the velocity gradients in 3D-extruded geometries which were previously computed with 2-point formulas. These computations illustrate the versatility of the diamond cell technique, which can be applied to other cut-cell methods.

Implementation and performance analysis of the massively parallel method of characteristics based on GPU

The method of characteristics (MOC) is one of the most common methods for solving the neutron transport equation in practical application. Researches have been focused on the acceleration techniques and the parallel algorithm for improving the efficiency of MOC. The Graphics Processing Unit (GPU) provides an alternative method of parallelizing the MOC neutron transport sweep. In this work, a GPU-paralleled 2D MOC code is implemented, which employs the diamond difference (DD) scheme and the step characteristics (SC) scheme. Different parallel schemes which are ray-level, energy-group-level, and polar-angle-level, are analyzed to choose the proper parallel scheme. The C5G7 2D benchmark is calculated to verify the accuracy and efficiency of the code in different schemes with single precision and double precision. The bottlenecks of the GPU code are identified and the code is classified into three categories, which are compute-bound, memory-bound, and latency-bound, according to the performance analysis model introduced in this paper. In addition, corresponding optimization strategies are applied to improve the performance according to the analysis. Moreover, the speed, power efficiency, and hardware cost are compared for CPU and GPU based on a fictitious quarter core PWR problem. Numerical results demonstrate that the energy group-level parallelization can obtain the optimal performance on GPU. Optimization strategies are effective to improve the efficiency of the calculation on GPU, which indicates that the performance analysis model is useful and effective to locate the limitation of the code. Moreover, the GPU-version code is about 30 times faster than the CPU-version code with double precision and about 100 times faster with single precision, while the desired accuracy is kept. And the GPU delivers superior performance in both speed, energy efficiency, and hardware cost.

Modal methods for the neutron diffusion equation using different spatial modes

The behaviour of the neutrons inside a nuclear reactor core can be modelled by using the time dependent neutron diffusion equation. Different time schemes have been used to integrate this equation. One possibility is to use a modal method, which is based on the expansion of the neutron flux in terms of spatial modes that are the eigenfunctions associated with a given configuration of the reactor core. Several spatial modes can be defined for the neutron diffusion equation such as the λ, α and γ-modes. In this work, the λ, the α and the γ-modes have been used to develop different modal kinetics equations, using a high order finite element method for the spatial discretization of the neutron diffusion equation. The performance of the different modal kinetic equations has been tested and compared using two 3D transient benchmark problems.

Three-dimensional exact hygro-thermo-elastic solutions for multilayered plates: Composite laminates, fibre metal laminates and sandwich plates

Three-dimensional (3D) exact hygrothermal elasticity solutions are developed to study the behaviour of simply supported rectangular multilayered composite plates, considering any combination of orthotropic and/or isotropic layers, under hygro-thermo-mechanical loadings. In agreement with the 3D exact elasticity solutions derived by Pagano in the early 1970s, the form of the through-thickness exact solutions of any given layer depends, in fact, on whether the layer material is isotropic or (at most) orthotropic. The 3D exact thermoelastic solutions found, already available, consider either only isotropic layers or only (at most) orthotropic composite layers. This work aims to be all-inclusive, in addition to a further development by considering hygrothermal elasticity. The results here presented consider composite laminates, fibre metal laminates and sandwich plates with different side-to-thickness ratios under a series of hygro-thermo-mechanical loadings. A transverse mechanical load and/or thermal load and/or hygroscopic load are imposed on the multilayered plate top and bottom surfaces to study the effects of hygrothermal environments in the multilayered plate behaviour. These effects are demonstrated by the through-thickness distributions of displacements, stresses, temperature and weight percent moisture content for the selected multilayered plates under different hygro-thermo-mechanical loadings, which may serve as 3D benchmark exact solutions.

A computational methodology for simulating quasi-brittle fracture problems

The paper focuses on an efficient and simple methodologies for simulating the three dimensional (3D) quasi-brittle fracture problems. Strain-softening is performed on the elements by a developed anisotropic continuum damage model that has more effective capability in crack path prediction and is easily available in standard finite elements. In the present damage model, the damaged stiffness tensor is constructed to form a crack surface, and the energy dissipation in the damaged element is only allowed in the direction perpendicular to the crack plane. Crack surface is divided into crack lines and crack triangles based on the first introduced crack surface discretization, and the application scope of local tracking algorithm is extended from two dimension to 3D. The present tracking algorithm not only guarantees the continuity and stability of the predicted crack path by solving the topological problems but also has low computational cost, keeping the advantages of local tracking. The method does not identify the crack plane within each element, but it couples well with smeared crack method by identifying all the elements through which the crack surface passes. The high efficiency and stability of the present approach are verified by resolving several 3D benchmark problems in failure analysis.

A GPU-enabled implicit Finite Volume solver for the ideal two-fluid plasma model on unstructured grids

This paper describes the main features of a pioneering unsteady solver for simulating ideal two-fluid plasmas on unstructured grids, taking profit of GPGPU (General-purpose computing on graphics processing units). The code, which has been implemented within the open source COOLFluiD platform, is implicit, second-order in time and space, relying upon a Finite Volume method for the spatial discretization and a three-point backward Euler for the time integration. In particular, the convective fluxes are computed by a multi-fluid version of the AUSM+up scheme for the plasma equations, in combination with a modified Rusanov scheme with tunable dissipation for the Maxwell equations. Source terms are integrated with a one-point rule, using the cell-centered value. Some critical aspects of the porting to GPU’s are discussed, as well as the performance of two open source linear system solvers (i.e. PETSc, PARALUTION). The code design allows for computing both flux and source terms on the GPU along with their Jacobian, giving a noticeable decrease in the computational time in comparison with the original CPU-based solver. The code has been tested in a wide range of mesh sizes and in three different systems, each one with a different GPU. The increased performance (up to 14x) is demonstrated in two representative 2D benchmarks: propagation of circularly polarized waves and the more challenging Geospace Environmental Modeling (GEM) magnetic reconnection challenge.

Unsplit perfectly matched layer absorbing boundary conditions for second-order poroelastic wave equations

An unsplit perfectly matched layer (PML), an absorbing boundary condition, is proposed for the second-order Biot’s equations to simulate wave propagation in unbounded poroelastic media. The PML formulas are derived using a two-parameter complex stretching approach, resulting in a two-dimensional (2D) time-domain system which consists of four second-order displacement equations and six first-order auxiliary differential equations. The proposed PML was validated using three 2D benchmarks: (i) spurious reflection of PML at grazing incidence; (ii) application of PML in stratified porous media and (iii) the relationship between the material property and the stability of PML. The numerical results show that the outgoing body waves and surface waves can be effectively attenuated by the proposed PML with appropriate scaling factors and damping profiles. The proposed formulas are much more compact, making it more efficient in studying poroelastic wave phenomena in open domain.

Discrete Boltzmann Equation model of polydisperse shallow granular flows

A polydisperse shallow mixture consisting of Np solid phases (Np ≥ 1) and one fluid phase (the ambient fluid) is a reliable model for several industrial and environmental flows, as e.g. landslides, avalanches, debris flows and fluidized beds. The description and prediction of these flows is of primary importance, mainly with respect to the mitigation and protection from natural hazards. This paper is aimed at deriving the polydisperse shallow granular flow equations by depth-averaging mass and momentum equations of the mixture and at formulating an equivalent Discrete Boltzmann Equation (hereinafter DBE) model as solution method. The reason is the simplicity and the versatility of the DBE, which consists of a set of purely advective, linear, first order partial differential equations, whose numerical integration does not need sophisticated methods. Both 1D and 2D benchmarks, concerning with the propagation of discontinuities in three-phase shallow granular flows, are obtained by applying the finite differences Lax-Friedrichs (hereinafter LF) method to the polydisperse shallow granular flow equations. The overall agreement is good, showing that the DBE and the LF numerical results are equivalent.

Classification of 3D Archaeological Objects Using Multi-View Curvature Structure Signatures

We propose a generalized 3D shape descriptor for the efficient classification of 3D archaeological artifacts. Our descriptor is based on a multi-view approach of curvature features, consisting of the following steps: pose normalization of 3D models, local curvature descriptor calculation, construction of 3D shape descriptor using the multi-view approach and curvature maps, and dimensionality reduction by random projections. We generate two descriptors from two different paradigms: 1) handcrafted, wherein the descriptor is manually designed for object feature extraction, and directly passed on to the classifier and 2) machine learnt, in which the descriptor automatically learns the object features through a pretrained deep neural network model (VGG-16) for transfer learning and passed on to the classifier. These descriptors are applied to two different archaeological datasets: 1) non-public Mexican dataset, represented by a collection of 963 3D archaeological objects from the Templo Mayor Museum in Mexico City that includes anthropomorphic sculptures, figurines, masks, ceramic vessels, and musical instruments; and 2) 3D pottery content-based retrieval benchmark dataset, consisting of 411 objects. Once the multi-view descriptors are obtained, we evaluate their effectiveness by using the following object classification schemes: $K$ -nearest neighbor, support vector machine, and structured support vector machine. Our object descriptors classification results are compared against five popular 3D descriptors in the literature, namely, rotation invariant spherical harmonic, histogram of spherical orientations, signature of histograms of orientations, symmetry descriptor, and reflective symmetry descriptor. Experimentally, we were able to verify that our machine learnt and handcrafted descriptors offer the best classification accuracy (20% better on average than comparative descriptors), independently of the classification methods. Our proposed descriptors are able to capture sufficient information to discern among different classes, concluding that it adequately characterizes the datasets.

MOANA: An Online Learned Adaptive Appearance Model for Robust Multiple Object Tracking in 3D

Multiple object tracking has been a challenging field, mainly due to noisy detection sets and identity switch caused by occlusion and similar appearance among nearby targets. Previous works rely on appearance models built on individual or several selected frames for the comparison of features, but they cannot encode long-term appearance changes caused by pose, viewing angle and lighting conditions. In this work, we propose an adaptive model that learns online a relatively long-term appearance change of each target. The proposed model is compatible with any feature of fixed dimension or their combination, whose learning rates are dynamically controlled by adaptive update and spatial weighting schemes. To handle occlusion and nearby objects sharing similar appearance, we also design cross-matching and re-identification schemes based on the application of the proposed adaptive appearance models. Additionally, the 3D geometry information is effectively incorporated in our formulation for data association. The proposed method outperforms all the state-of-the-art on the MOTChallenge 3D benchmark and achieves real-time computation with only a standard desktop CPU. It has also shown superior performance over the state-of-the-art on the 2D benchmark of MOTChallenge.

Maximum length scale requirement in a topology optimisation method based on NURBS hyper-surfaces

This paper deals with a new method for handling manufacturing and geometrical requirements in the framework of a general Topology Optimisation (TO) strategy. In particular, the maximum length scale constraint (MLSC) implementation is addressed in order to obtain multiple load paths or to locally limit the size of the component. The classic formulation of the MLSC is revisited in the framework of a density-based TO algorithm wherein the pseudo-density field is represented through a NURBS hyper-surface. The NURBS hyper-surface properties are exploited to effectively formulate the MLSC. The effectiveness of the proposed approach is proven on a meaningful 3D benchmark.

Combined length scale and overhang angle control in minimum compliance topology optimization for additive manufacturing

This paper focuses on topology optimization for additive manufacturing. In order to ensure that the optimized design is immediately manufacturable, it is essential to take into account the appropriate geometric constraints during the optimization. Two important constraints are minimum length scale and maximum overhang angle. A minimum length scale is needed to ensure that the condition on minimal printable feature sizes is satisfied, while an imposed overhang angle eliminates the need for a temporary support structure. This paper first shows that both constraints cannot simultaneously be met by a straightforward coupling of existing methods for length scale and overhang angle control. Next, a new filtering scheme is introduced, based on a specific combination of spatial filters, which allows direct control over these constraints in a minimum compliance topology optimization problem. A 2D benchmark problem and a complex 3D case study are presented to demonstrate that the proposed filtering scheme successfully imposes a target length scale in both the solid and the void phase of the design domain, while simultaneously allowing control over the overhang angle.