Logo. A Massively-Parallel Multicore Acceleration of a Point Contact Solid Mechanics Simulation DRAFT

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1 Paper 1 Logo Civil-Comp Press, 2017 Proceedings of the Fifth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, P. Iványi, B.H.V Topping and G. Várady (Editors) Civil-Comp Press, Stirlingshire, Scotland A Massively-Parallel Multicore Acceleration of a Point Contact Solid Mechanics Simulation M. Kolman, G. Kosec Parallel and Distributed Systems Laboratory, Jožef Stefan Institute, Ljubljana, Slovenia Abstract This paper deals with the numerical determination of the stress and displacement distribution in a solid body subjected to the applied external force. The tackled solid mechanics problem is governed by the Navier-Cauchy equation that describes the deformation within the solid body through the displacement vector field. To obtain the solution, a coupled system of non-linear Partial Differential Equations (PDE) of second order has to be solved. In this paper, the problem is approached by a strong form Moving Least Squares (MLS) based numerical discretization also referred to as a Meshless Local Strong Form Method (MLSM). A generic C++ implementation of a MLSM is used for demonstration of parallel solution of a Point Contact problem on Intel Xeon Phi multicore accelerator. All tests are executed on either the host machine with two Intel Xeon E v3 6 core processors or offloaded to its 60 core Intel Xeon Phi SE10/7120 series. The shared memory parallelization is implemented through an OpenMP API. Keywords: MLSM, meshless, OpenMP, Intel Xeon Phi coprocessor, parallel implementation 1 Introduction In the majority of numerical simulations the solid mechanics problems are tackled with Finite Elements Methods (FEM) [1]. However, as an alternative to the mesh based FEM, a new class of numerical methods, referred to as Meshless methods, has emerged to alleviate the meshing complexity, which is in many cases, especially in 3D, still the most cumbersome part of the solution procedure. Both of the weak and strong form variants of the Meshless methods have been already applied on the solid mechanics problems [2, 3]. 1

2 In this paper we consider a Strong Form Meshless method that is based on the MLS approximation [4]. The most important features of the employed numerical method are the locality and generality. The locality is manifested by the fact that the evaluation of partial differential operators, which is the main part of the solution process, relies only on a small number of surrounding computational nodes. This is important from the computation point of view, since it reduces the inter-processor communication, which is often the bottleneck of parallel algorithms [5]. The generality, on the other hand, arises from the fact that all the building blocks of the method depend only on the distance between the computational nodes. This is a very useful feature, especially when dealing with problems in multidimensional spaces, complex geometries, and moving boundaries. The generality of the method also allows for an elegant computer implementation. All these facts make the MLSM an attractive alternative to the classical approaches like Finite Differences Method that suffers from geometrical limitations, and more general weak form Finite Elements Method that require meshing. 2 Point contact problem The Navier-Cauchy equations describe the dynamics of a solid through the displacement vector field u, expressed concisely in vector form as ρ 2 u t = (λ + µ) ( u) + 2 µ 2 u D u (1) t where µ and λ stand for Lamé constants, ρ is the density, and D is damping coefficient. The point contact problem on the lower half-plane, i.e. a point-load applied to a surface of the body, is characterised with the traction boundary conditions σ xy (x, 0) = 0, σ yy (x, y) = P δ(x, y) (2) where σ xy and σ xx are shear and normal stress tensor components, and δ(x, y) is the Dirac delta function. These boundary conditions state that there is a singular normal force P applied at (x, y) = (0, 0) and that there are no shear stresses on the surface of the elastic half- plane. The problem s solution can be expressed in a closed form and is therefore ideal for testing purposes. The solution is presented as σ xx = 2P π x 2 y (x 2 + y 2 ) 2, σ yy = 2P π y 3 (x 2 + y 2 ) 2, σ xy = 2P π xy 2 (x 2 + y 2 ) 2, (3) for a point (x, y) in the half-plane. For our purposes a solution in terms of displacement vector u = (u x, u y ) is more convenient. It can be obtained by integrating Eq. (3), which yields u x = P ( 4πµ u y = P 4πµ (κ 1)θ 2xy ), (4) r ((κ 2 ) + 1) log r + 2x2, (5) r 2 2

3 where r = x 2 + y 2, tan θ = x. and κ = 3 4ν, with ν standing for the Poisson y ratio. 3 Meshless Local Strong Form numerical approach To numerically solve a problem at hand, we discretize a domain into a finite set of nodes, at which the partial differential operators, occurring in Eq. (1), are evaluated. The core concept of the spatial discretization used here, namely a MLSM, is a local Moving Least Squares (MLS) approximation of a considered field over the overlapping local support domains. In each node we create an approximation function over a small local sub-set of neighbouring n nodes acting as a trial function û(p) = m α i b i (p) = b(p) T α, (6) i=1 with m, α, b, p (x, y) standing for the number of basis functions, approximation coefficients, basis functions and the position vector, respectively. The problem can be written in matrix form as α = ( W 0.5 B ) + W 0.5 u, (7) where (W 0.5 B) + stand for a Moore Penrose pseudo inverse. By explicitly expressing the coefficients α into the trial function one gets û (p) = b(p) T( W 0.5 (p) B ) + W 0.5 (p) u = χ (p) u, (8) where χ stand for the shape functions. Now, we can apply any partial differential operator L, as is our goal, on the trial function L û (p) = Lχ (p) u. (9) The presented formulation is convenient for implementation since most of the complex operations, i.e. finding support nodes and building shape functions, are performed only when nodal topology changes. During computation the pre- computed shape functions are convoluted with the vector of field values in the support to evaluate the desired operator. With the explicit temporal discretization the solution of Eq. (1) is formulated as u 3 = t2 ρ ( (λ + µ) ( u 2 ) + µ 2 u 2 D u ) 2 u 1, (10) t where u 1, u 2 and u 3 stand for displacement vector at two previous time steps and the one currently computed, respectively. The operators L =,, 2 are computed with Equations (8) and (9) [4]. 3

4 4 Implementation A MLSM solution for the point contact problem is implemented in C++ and compiled either with GNU C++ compiler g or Intel s C++ compiler ICPC version (which has GCC version compatibility) both with compiler flags -O3 -std=c++14 -fopenmp. The code is developed within an open source project and is freely available at [6]. It implements different domain classes, a Nearest Neighbour (knn) search based on the kd-tree algorithm, a general Moving Least Squares (MLS) engine and its extension to a full MLSM engine. 4.1 Building the shape functions In a first major step all the necessary shape functions needed to calculate the required differential operator Lû(p) are prepared. This set-up procedure comprises positioning of the nodes, finding support nodes, defining the MLS approximation, and finally creating the operators class. The implementation concept is schematically presented in Figure 1, and written in a C++ with MLSM library [6] as 1 RectangleDomain <Vec2d > domain ( domain_lo, domain_hi ) ; 2 domain. f i l l U n i f o r m I n t e r i o r W i t h S t e p ( d_space ) ; domain. f i l l U n i f o r m B o u n d a r y W i t h S t e p ( d_space ) ; 4 domain. f i n d S u p p o r t ( n ) ; s u p p o r t = domain. p o s i t i o n s [ domain. s u p p o r t [ 0 ] ] ; 6 EngineMLS<Vec2d, Gaussians, Gaussians > mls ( { sigmab, m}, s u p p o r t, sigmaw ) ) ; 8 auto mlsm = make_mlsm ( domain, mls ) ; The domain is an object containing all information about the nodes, including support domain of all nodes. It is formed by parameters domain_lo, domain_hi and d_space, which define its boundaries and node density. The variable mlsm is an instance of the MLSM class that is capable of calculating certain differential operators (e.g. Laplacian, grad, div, etc.) of fields defined on the domain. The variables sigmab and sigmaw are the standard deviations of basis functions and weight function, respectively, both Gaussians in present case, defined as g(p) = 1 ) exp ( p 2, (11) 2πσ 2σ The m defines the number of basis functions and n the number of support nodes. In this paper the same number of basis functions and support nodes is assumed. The n = m = 9 assumption effectively reduces the approximation to the collocation, which is a popular set-up in a meshless community [7, 8], especially when regular nodal distributions are used. However, in cases when irregular distributions are needed the overdetermined MLS is preferred [4]. 4

5 4.2 Time simulation Figure 1: A MLSM implementation diagram. Before the time stepping the boundary conditions and initial state are set. The boundaries are assumed to be of a Dirichlet type with values obtained from the closed form solution, while the initial state of the displacement is set to zero throughout whole the domain, implemented as 1 Range < vec_t > u_1 ( domain. s i z e ( ), 0 ), 2 u_2 ( domain. s i z e ( ), 0 ), u_3 ( domain. s i z e ( ), 0 ) ; 4 u_3 [ boundary ] = u _ a n a l y t i c a l ( boundary ) ; Note, that the boundary is a vector of indices of boundary nodes. With prepared operators, known two consequential previous time steps and the boundary conditions, the Eq. (10) can be numerically solved as 1 #pragma omp p a r a l l e l f o r p r i v a t e ( j ) s c h e d u l e ( s t a t i c ) 2 f o r ( j = 0 ; j < N; ++ j ) { u_3 [ i ] = d t * d t / rho * ( 4 mu * mlsm. l a p ( u_2, i ) + E / (2 2 * nu ) * mlsm. g r a d d i v ( u_2, i ) 6 D * ( u_2 [ i ] u_1 [ i ] ) / d t ) + 2 * u_2 [ i ] u_1 [ i ] ; 8 } The N states the number of nodes in the domain and dt is the size of the time step. Variables mu, rho, E and nu are physical constants of the material corresponding to the first Lamé constant µ, density ρ, Young modulus E and Poisson coefficient ν, respectively. The variable mlsm is an operator object holding shape functions and procedures for computing the partial differential operators. Since the loop iterations are independent the whole process can be easily parallelized with OpenMP API by using a compiler option #pragma omp. Once the u 3 is computed for all nodes in the domain, the step forward can be 5

6 performed simply by u 1 = u 2 and u 2 = u 3. This iterative process takes place until the steady state is achieved. 5 Results 5.1 Solution of a point contact problem The point contact problem is solved on a domain (x, y) Ω = [ 1, 1] [ 1, 0.01]. The standard deviation of basis functions is 70 times the domain characteristic distance, i.e. the average distance to the closest neighbour. The physical constants are set to ρ = 7874 [kg/m 3 ], ν = 0.25, E = [Pa], D = 10 9 and the applied force is P = 1000 [Pa]. All calculations are done with dt = 10 7 [s]. In Figure 2 a MLSM solution û(p) computed with N = regularly distributed nodes is presented and compared against the known analytical solution u(p), with relative displacement error computed as E = û(p) u(p) / ( u(p) ) (12) and visualized through the color map. The displacement field has been, for the sake of visibility, multiplied with a factor of Figure 2: Relative displacement error E displayed on displaced nodes, with magnified displacement by a factor of Next comparison is focused on the displacement over two horizontal crosssections, at y = and y = 0.5. The numerical and analytical solutions are presented in Figure 3 and it is evident that MLSM agrees well with a known solution. 6

7 uy(x,y = 0.015),ux(x,y = 0.015) u y MLSM u y analytical -5 u x MLSM u x analytical x uy(x,y = 0.5),ux(x,y = 0.5) x (a) Horizontal displacement cross section at the (b) Horizontal displacement cross section in top of the domain. the middle of the domain u y MLSM u y analytical u x MLSM u x analytical Figure 3: Two horizontal crosssections of the analytical and numerical solution for the point contact problem. 5.2 Execution performance We tested the execution on a 60-core computing machine Intel Xeon Phi Coprocessor SE10/7120 series and its server host with two Intel Xeon CPU E v3 processors. The two machines are compared in Table 1. Name Cores Clock [GHz] L2 Cache [MB] Memory [GB] Intel Xeon Phi (0.5/core) 16 Intel Xeon CPU (2.5/core) 64 Table 1: Specification comparison between Intel Xeon Phi and Intel Xeon CPU. In this preliminary study we are interested in the scalability of the execution performance. In Figures 4 and 5 a speed-up, defined as S = T (1) T (N t ), (13) with T (N t ) standing for execution time on N t threads, for different domain sizes and number of utilized threads is presented. The domain size is ranged from N = 741 up to N = nodes. For both computers all available threads, including hyperthreading, are utilized. On the host machine in total 12 physical cores support up to 24 threads and on the coprocessor 120 threads can be used on its 60 physical cores. 7

8 S N=741 N=2178 N= N=79401 N= N= N= N= N t (a) Execution speedup on the Intel Xeon (b) Execution speedup on Intel host. Coprocessor. S N=741 N=2178 N=19701 N=79401 N= N= N= N= N t Xeon Phi Figure 4: Shared memory parallelization speedup with respect to the number of threads for different problem sizes. Increasing the number of threads on the host is beneficial only up to the limit of physical cores, while on the coprocessor using additional 2 threads also improves results. It can be also seen that coprocessor requires much bigger problems to show its full potential in terms of scalability. The maximal efficiency (S/N t ) of 0.6 is achieved on host already on a relatively small systems (N = 10 5 ), while the same efficiency is achieved on the coprocessor only with systems consisting of N = nodes. 6 Conclusions In this paper we demonstrated the application of the MLSM discretization technique on the solution of a coupled system of second order partial differential equations. Namely we solved a Navier-Cauchy equation that describes displacements in a solid body subjected to an external force. The code has been written in C++ and executed on two different computing architectures, i.e. the Intel Xeon server class CPU and Intel Xeon Phi coprocessor. On both architectures a relatively good scalability has been achieved, with a maximal parallel efficiency of 0.67 for 12 cores on a host machine and 60 cores on a coprocessor. Although, the parallel efficiency on the coprocessor is good, the overall performance is not satisfactory. It is our main focus in future work to improve the performance of the MLSM on the coprocessor by means of improving the utilization of the vectorization in the lowest level operations, namely the convolution of the shape 8

9 S N t = 1 N t = 2 3 N t = 4 N t = 8 2 N t = 12 N t = 16 N t = N N (a) Execution speedup on the Intel Xeon (b) Execution speedup on Intel host. Coprocessor. S N t = 1 N t = 2 N t = 5 N t = 10 N t = 30 N t = 60 N t = 120 Xeon Phi Figure 5: Shared memory parallelization speedup with respect to the problem size for different number of utilized threads. functions and the field values in the support nodes. Acknowledgement The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0095) References [1] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method: Solid Mechanics, Butterworth-Heinemann, [2] Y. Chen, J.D. Lee, A. Eskandarian, Meshless methods in solid mechanics, Springer, New York, NY, [3] B. Mavrič, B. Šarler, Local radial basis function collocation method for linear thermoelasticity in two dimensions, International Journal of Numerical Methods for Heat and Fluid Flow, 25: , [4] G. Kosec, A local numerical solution of a fluid-flow problem on an irregular domain, Advances in Engineering Software, in press,

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