FPGA architecture and implementation of sparse matrix-vector multiplication for the finite element method

The Finite Element Method (FEM) is a computationally intensive scientific and engineering analysis tool that has diverse applications ranging from structural engineering to electromagnetic simulation. The trends in floating-point performance are moving in favor of Field-Programmable Gate Arrays (FPGAs), hence increasing interest has grown in the scientific community to exploit this technology. We present an architecture and implementation of an FPGA-based sparse matrix-vector multiplier (SMVM) for use in the iterative solution of large, sparse systems of equations arising from FEM applications. FEM matrices display specific sparsity patterns that can be exploited to improve the efficiency of hardware designs. Our architecture exploits FEM matrix sparsity structure to achieve a balance between performance and hardware resource requirements by relying on external SDRAM for data storage while utilizing the FPGAs computational resources in a stream-through systolic approach. The architecture is based on a pipelined linear array of processing elements (PEs) coupled with a hardware-oriented matrix striping algorithm and a partitioning scheme which enables it to process arbitrarily big matrices without changing the number of PEs in the architecture. Therefore, this architecture is only limited by the amount of external RAM available to the FPGA. The implemented SMVM-pipeline prototype contains 8 PEs and is clocked at 110 MHz obtaining a peak performance of 1.76 GFLOPS. For 8 GB/s of memory bandwidth typical of recent FPGA systems, this architecture can achieve 1.5 GFLOPS sustained performance. Using multiple instances of the pipeline, linear scaling of the peak and sustained performance can be achieved. Our stream-through architecture provides the added advantage of enabling an iterative implementation of the SMVM computation required by iterative solution techniques such as the conjugate gradient method, avoiding initialization time due to data loading and setup inside the FPGA internal memory.     

Non-Newtonian effects of blood flow in complete coronary and femoral bypasses

A numerical investigation of non-Newtonian steady blood flow in a complete idealized 3D bypass model with occluded native artery is presented in order to study the non-Newtonian effects for two different sets of physiological parameters (artery diameter and inlet Reynolds number), which correspond to average coronary and femoral native arteries. Considering the blood to be a generalized Newtonian fluid, the shear-dependent viscosity is evaluated using the Carreau–Yasuda model. All numerical simulations are performed by an incompressible Navier–Stokes solver developed by the authors, which is based on the pseudo-compressibility approach and the cell-centred finite volume method defined on unstructured hexahedral computational grid. For the time integration, the fourth-stage Runge–Kutta algorithm is used. The analysis of numerical results obtained for the non-Newtonian and Newtonian flows through the coronary and femoral bypasses is focused on the distribution of velocity and wall shear stress in the entire length of the computational model, which consists of the proximal and distal native artery and the connected end-to-side bypass graft.     

Stochastic optimization for the calculation of the optimal critical curve from experimental data in a model of the process of regaining balance after perturbation from quiet stance

We demonstrate the successful application of ALOPEX stochastic optimization to the problem of calculating the optimal critical curve in a dynamical systems model of the process of regaining balance after perturbation from quiet stance. Experimental data provide the time series of angles for which the subjects were able to regain balance after an initial perturbation. The optimal critical curve encloses all data points and has a minimum distance from the border points of the data set. We demonstrate the results of the optimization firstly using the traditional cost function of chi-square distance. We then successfully introduce a modified cost function that fits the model to the experimental data by taking into account the specific requirements of the model. By use of the proposed cost function, combined with the efficiency of our optimization method, an optimal critical curve is calculated even in the cases of very asymmetric data sets that lie within the capabilities of the existing model.     

WEB-Spline-based mesh-free finite element approximation for p-Laplacian

In this paper, we discuss the approximation of p-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of the p-Laplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations.     

Stochastic optimization for the calculation of the time dependency of the physiological demand during exercise and recovery

The stochastic optimization method ALOPEX IV is successfully applied to the problem of estimating the time dependency of the physiological demand in response to exercise. This is a fundamental and unsolved problem in the area of exercise physiology, where the lack of appropriate tools and techniques forces the assumption and the use of a constant demand during exercise. By the use of an appropriate partition of the physiological time series and by means of stochastic optimization, the time dependency of the physiological demand during heavy intensity exercise and its subsequent recovery is, for the first time, revealed.     

Stochastic optimization for modeling physiological time series: application to the heart rate response to exercise

Stochastic optimization is applied to the problem of optimizing the fit of a model to the time series of raw physiological (heart rate) data. The physiological response to exercise has been recently modeled as a dynamical system. Fitting the model to a set of raw physiological time series data is, however, not a trivial task. For this reason and in order to calculate the optimal values of the parameters of the model, the present study implements the powerful stochastic optimization method ALOPEX IV, an algorithm that has been proven to be fast, effective and easy to implement. The optimal parameters of the model, calculated by the optimization method for the particular athlete, are very important as they characterize the athlete's current condition. The present study applies the ALOPEX IV stochastic optimization to the modeling of a set of heart rate time series data corresponding to different exercises of constant intensity. An analysis of the optimization algorithm, together with an analytic proof of its convergence (in the absence of noise), is also presented.     

On finite element approximation of general boundary value problems in nonlinear elasticity

This paper deals with the approximate solution of the general boundary value problem in nonlinear elasticity by the finite element displacement method. Under usual conditions which also guarantee the existence of locally unique solutions the quasi-optimal convergence inL ² andL ^∞ is shown for displacement fields and stresses. Furthermore a projective Newton method is considered which reduces the solution of the nonlinear continuous problem to the successive solution of a sequence of linearized problems of increasing dimension. It is proved that this procedure is well defined and also converges with quasi-optimal rates.     

Three-dimensional spectral element simulations of variable density and viscosity, miscible displacements in a capillary tube

A high-accuracy numerical approach is introduced for three-dimensional, time-dependent simulations of variable density and viscosity, miscible flows in a circular tube. Towards this end, the conservation equations are treated in cylindrical coordinates. The spatial discretization is based on a mixed spectral element/Fourier spectral scheme, with careful treatment of the singularity at the axis. For the temporal discretization, an efficient semi-implicit method is applied to the variable viscosity momentum equation. This approach results in a constant coefficient Helmholtz equation, which is solved by a fast diagonalization method. Numerical validation data are presented, and simulations are conducted for the three-dimensionally evolving instability resulting from an unstable density stratification in a vertical tube. Some preliminary comparisons with corresponding experiments are undertaken.     

A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains

A finite volume scheme which is based on fourth order accurate central differences in the spatial directions and on a hybrid explicit/semi-implicit time stepping scheme was developed to solve the incompressible Navier-Stokes equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity at the singularity of the cylindrical coordinate system and a new stability condition. The new method was applied in the direct numerical simulations (DNS) of the fully developed non-swirling turbulent flow through straight pipes with circular cross-section for the Reynolds number Re_τ = 360 based on the friction velocity u_τ and the pipe diameter. The obtained results are expressed in terms of statistical moments of the velocity components and are presented in comparison with those obtained with a second order accurate scheme and by measurements. It is shown that the fourth order spatial discretization leads to improved higher order statistical moments, while the first and the second order moments are more or less insensitive to the spatial discretization order.     

On the validity of an approximation available in the finite element shell analysis

A simplified displacement method is proposed for the finite element shell analysis. This method requires two kinds of shape functions in the evaluation of the membrane and bending stiffness of the shell to curtail the computational processes of the usual displacement method. Mathematical and numerical studies are also made to establish the validity of the approach.     

Nonconforming finite element approximation of the Giesekus model for polymer flows

We present a numerical approximation of the Giesekus equation which is considered as a realistic model for polymer flows. We use nonconforming finite elements on quadrilateral grids which necessitate the addition of two stabilization terms. An appropriate upwind scheme is employed for the convective term. The underlying discrete Stokes problem is then analysed. Finally, numerical tests are presented in order to validate the code, illustrating its good behavior for large Weissenberg numbers. Comparisons with Polyflow® and with the literature are also carried out.     

Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations

Poor performances can be obtained from classical domain decomposition algorithms to solve advection-diffusion equations in the case of convection dominated flows. Therefore, adaptive domain decomposition have been developed for such flows. We investigate the properties of some algorithms of this kind in the framework of a finite volume/finite element discretization.This research was carried out while the author was visiting the Group of Applied Mathematics and Simulation of CRS4, and was supported by an HCM fellowship.     

Review of energy conservation errors in finite element softwares caused by using energy-inconsistent objective stress rates

The paper briefly summarizes the theoretical derivation of the objective stress rates that are work-conjugate to various finite strain tensors, and then briefly reviews several practical examples demonstrating large errors that can be used by energy inconsistent stress rates. It is concluded that the software makers should switch to the Truesdell objective stress rate, which is work-conjugate to Green’s Lagrangian finite strain tensor. The Jaumann rate of Cauchy stress and the Green-Naghdi rate, currently used in most software, should be abandoned since they are not work-conjugate to any finite strain tensor. The Jaumann rate of Kirchhoff stress is work-conjugate to the Hencky logarithmic strain tensor but, because of an energy inconsistency in the work of initial stresses, can lead to severe errors in the cases of high natural orthotropy or strain-induced incremental orthotropy due to material damage. If the commercial softwares are not revised, the user still can make in the user’s implicit or explicit material subroutines (such as UMAT and VUMAT in ABAQUS) a simple transformation of the incremental constitutive relation to the Truesdell rate, and the commercial software then delivers energy consistent results.     

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.     

Dual iterative techniques for solving a finite element approximation of the biharmonic equation

A finite element approximation of the Dirichlet problem for the biharmonic operator is described. Its main feature is that it is equivalent to solving a sequence of discrete Dirichlet problems for the operator -Δ. This method, which has already been shown to be convergent, is particularly well-suited for problems in fluid dynamics.     

A stabilized finite element method for modeling of gas discharges

An efficient stabilized finite element method for modeling of gas discharge plasmas is represented which provides wiggle-free solutions without introducing much artificial diffusion. The stabilization is achieved by modifying the standard Galerkin test functions by means of a weighted quadratic term that results in a consistent Petrov-Galerkin formulation of the charge carriers in the plasma. Using the example of a glow discharge plasma in argon, it is shown that this efficient method provides more accurate results on the same spatial grid than the widely used finite difference approach proposed by Scharfetter-Gummel if the weighting factor is determined in dependence on the local Péclet number and the modified test functions are consistently applied to all terms of the governing equations.     

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.     

On the density correlation of the spontaneous fluctuation in a real-coded lattice gas

We investigate the spontaneous fluctuation in a real-coded lattice gas (RLG) model by studying the density correlation function. In particular, the dynamic structure factor obtained from RLG is in agreement with the Rayleigh-Brillouin spectrum, a spectrum which can also be measured from a real fluid at the continuum limit. We also work out the analytic form of the static structure factor for the RLG model, which is supported by the numerical results.