SELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND

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1 Student Submission for the 5 th OpenFOAM User Conference 2017, Wiesbaden - Germany: SELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND TESSA UROIĆ Faculty of Mechanical Engineering and Naval Architecture, Ivana Lučića 5, Zagreb, Croatia, tessa.uroic@fsb.hr Keywords: multigrid, linear solvers, implicitly coupled systems, foam-extend Introduction Discretisation of partial differential equations (PDEs) in the framework of the Finite Volume Method (FVM) typically produces very large sparse matrices with a small number of nonzero entries. Most recent algorithms employ implicit methods in which all equations are solved simultaneously in a single block matrix, including variable cross-coupling terms. In many cases, there exists a non-linear connection between variables, which is eliminated by linearising the equations in order to avoid high-cost non-linear solvers. Choice of an appropriate solver is strongly related to the discretisation method, i.e. matrix properties. As mentioned before, the matrix resulting from the FVM discretisation is sparse, the number of unknowns is the same as the number of equations, thus the matrix is square. Its structure depends on the mesh numbering. Consistent mesh numbering in foam-extend ensures that the matrix is banded, i.e. the non-zero coefficients are grouped around the diagonal in a narrow stripe. Most linear solvers have the optimal performance for certain types of matrices: diagonally dominant, symmetric and positive-definite. Simple iterative solvers such as Jacobi and Gauss Seidel [1] operate in a point-by-point manner and incrementally improve the solution. Thus, in an N-dimensional space of an N by N matrix, fixed-point methods visit each direction of the N-dimensional space separately. On the other hand, Krylov subspace methods [1], such as Generalised Minimal Residual (GMRES) and Conjugate Gradient (CG) methods and its variants, choose the solution direction by evaluating the residual and searching for its smallest components. These methods are fairly efficient when combined with matrix preconditioning [1]. The most powerful class of iterative linear solvers are the multigrid methods [2]. They were originally created for systems of discretised elliptic PDEs but were later expanded and have proven to be efficient for general types of PDEs. Multigrid methods exploit the fact that the beforementioned point-fixed methods tend to quickly reduce the high frequency solution errors, i.e. the errors whose direction corresponds to the largest eigenvalues of the matrix. However, the low frequency errors remain and this is why the performance (convergence) of the fixed-point methods deteriorates. To solve this issue, multigrid methods construct a hierarchy of grids by coarsening the initial grid. The low frequency errors on the finer grid become high frequency errors on the coarser grid and the fixed-point algorithms are able to efficiently reduce these errors. The correction obtained on the coarser grid is then transferred back to the finer grid. This procedure of fine-to-coarse and coarse-to-fine grid communication can be repeated multiple times in various directions which will be defined by the multigrid cycle type. Algebraic multigrid methods operate on matrix coefficients directly and do not need a computational grid. Multigrid is typically used as an extremely efficient preconditioner with the Krylov subspace methods. In this abstract, a newly implemented algebraic multigrid method in foam-extend will be presented. The method is based on the classical multigrid coarsening algorithm (SAMG) [3].

2 Classical SAMG algorithm The new algorithm relies on the existing AMG algorithm and its functionalities including multigrid cycles, smoothers and fine level solvers, with a new method for choosing the coarse level and new coarse-to-fine and fine-to-coarse communication. Construction of the coarse level matrix relies on choosing the representative equations based on the following criteria: 1. In a single matrix row (equation), coefficients with sign opposite to diagonal coefficient are labeled as negative connections. All such coefficients larger than a fraction of the strongest value are strong connections called influences of the equation. Coefficients in the corresponding column of the diagonal coefficient are called dependencies of the equation. 2. Equations with the largest number of dependencies will become coarse. Their strongly dependent neighbours will become fine. 3. There cannot exist two coarse equations that strongly influence one another. 4. Every fine equation must have at least one strong coarse neighbour. However, this rule may be disregarded if there exists an equation without strong connections. After choosing the equations which will remain on the coarse level, interpolation for coarse-to-fine communication must be explicitly calculated. A prolongation matrix for interpolating the solution correction from coarse to fine level is calculated using the equation which describes the algebraically smooth error: a ii e i + j N i a ij e j = 0. (1) where a ii e i = j C i a ij e j j F s i a ij e j j F w i a ij e j, (2) N i is the subset of all neighbours of i C i is the subset of strong coarse neighbours of i F w i F s i is the subset of strong fine neighbours of i is the subset of weak fine neighbours of i The approximation of equation 1 yields interpolation weights for contribution from a single coarse equation to the fine equation: 1 w = α a ii + a C j F w a ij, (3) ij i where α is the scaling factor taking into account negative connections which are not coarse: j C i a ij + j F w = s a ij i > 1 (4) j C i a ij The restriction matrix for transferring the residual from fine to coarse level is obtained as a transpose of prolongation matrix. The coarse level matrix is obtained as a triple product of restriction, fine level matrix and prolongation: A coarse = R A fine P. (5) 2

SAMG for block-matrices For scalar matrices, previously described approach can be used directly, while for block matrices arising from implicitly coupled equations, a primary matrix must be chosen,

3 SAMG for block-matrices For scalar matrices, previously described approach can be used directly, while for block matrices arising from implicitly coupled equations, a primary matrix must be chosen, [4]. Interpolation formulas will be based on the primary matrix and the connectivity pattern of the primary variable. The first option is to choose the equations belonging to a single variable as the primary matrix. For example, in the case of implicitly coupled pressurevelocity solver [5], a logical choice for the primary matrix is the pressure equation which has favourable properties: it is elliptic, its matrix is symmetric and the sign of off-diagonal coefficients is opposite to the sign of diagonal coefficient. The coupling of the pressure and velocity is linear, thus the obtained interpolation formulas can be efficiently used for the momentum equation. The second option is to calculate the norm of each block coefficient and use the norm as a criterion in the coarsening procedure. The coarsening algorithm is applied on the chosen primary matrix. The definition of interpolation formulas for the variables of a point P k corresponds to the transfer of interpolation weights wk,l P, where l P k P to diagonal elements of W k,l. They are transferred to the w i,j with i P k and j P l such that i and j belong to the same unknown [4]. w P k,l 0 W k,l =..... (6) 0 wk,l P Another simplified method of choosing the coarse level matrix is the aggregation-based algebraic multigrid (AAMG), [6]. Instead of selecting certain equations for the coarse level, the fine level equations are joined to form aggregates (clusters). These aggregates are represented by a single equation on the coarse level. The interpolation operator does not have to be explicitly calculated as the values from the coarse level are directly copied (injected) for each member of the aggregate (with options of geometrical scaling). The coarse level matrix is formed by adding up the equations (additive correction, [7]). The implementation of this approach is straightforward but the convergence is limited by the incorrect interpolation formulas. In the next section, performance of the algorithm based on selection of coarse equations is compared to performance of the aggregation-based multigrid for the pressure-velocity block-coupled algorithm. Results The case chosen for examining the performance of SAMG is the 2D turbulent flow over a backward facing step, Fig. 1. The setup of the linear solver for the coupled velocity and pressure equation is shown in Tab. 1. First test case was run with the aggregative multigrid and the criterion for joining cells into aggregates was calculated using the pressure equation (component norm). The same norm (or primary matrix) was used for the selective multigrid. Selective multigrid was also tested with the Frobenius (two norm) which takes into account the momentum equation. For all cases, the minimal number of coarse level equations was 4, relative tolerance and ILUCp with fill in level 1 was used for smoothing the error. Figure 1: Pressure and velocity field for the backward facing step case. Different algorithms and norms have produced different coarse levels, which is illustated in Fig. 2. For SAMG, the cells are coloured by the spectrum (from blue to magenta) and the colour denotes the strength or 3

4 Table 1: Setup of AMG solver for the Up system Algorithm AAMG SAMG SAMG Cycle W-cycle W-cycle W-cycle Norm Component Component Frobenius Smoother ILUCp (fill in 1) ILUCp (fill in 1) ILUCp (fill in 1) Min. coarse equations Relative tolerance Number of coarsening levels Number of linear iterations Converged 111 s 50 s 117 s order of appearance of coarse equations. Blue cells have the strongest equation (they become coarse the first), while magenta is the weakest (they become coarse the last). This fact is very important because the order of appearance of equations will dictate the order of smoothing and will also affect the structure of the coarse level matrix. Figure 2: First coarsening level, from top to bottom: AAMG, SAMG with component norm, SAMG with Frobenius norm AAMG and SAMG have produced different number of coarsening levels: AAMG has 7 while SAMG has 5. The second and fifth level are shown in Figs. 3 and 4, respectively. Figure 3: Second coarsening level, from top to bottom: AAMG, SAMG with component norm, SAMG with Frobenius norm It is obvious that the two norms used with SAMG produce different coarse levels. In this case, the component norm, or the interpolation based on the pressure equation gave better convergence. The reason could be the interpolation formula itself or the efectiveness of the smoother on coarse level matrices with 4

5 Figure 4: Fifth coarsening level, from top to bottom: AAMG, SAMG with component norm, SAMG with Frobenius norm 20 Number of linear solver iterations AAMG (cluster) SAMG (two norm) SAMG (component norm) Residual 1e+00 1e-01 1e-02 1e-03 1e-04 1e-05 Backwardfacing step - pucoupledfoam - cluster U x U y U z p k ε 1e Outer iteration 1e Execution time 1e+00 1e-01 1e-02 Backwardfacing step - pucoupledfoam - SAMG - component norm U x U y U z p k ε 1e+00 1e-01 1e-02 Backwardfacing step - pucoupledfoam - SAMG - Frobenius norm U x U y U z p k ε Residual 1e-03 1e-04 Residual 1e-03 1e-04 1e-05 1e-05 1e-06 1e-06 1e Execution time 1e Execution time Figure 5: Number of linear iterations and convergence for all cases mentioned in Tab. 1. different structure. AAMG takes 870 linear iterations until convergence (tolerance is 10 6 in all cases) with time per iteration of 0.13 s. SAMG with component norm takes only 175 iterations and time per iteration is 0.29 s. SAMG with Frobenius norm takes 519 iterations or 0.22 s per iteration. Convergence and number of linear iterations are shown in Fig. 5. Conclusion 1. Selective algebraic multigrid (SAMG) converges faster than the agglomerative algebraic multigrid (AAMG) due to the better interpolation of algebraically smooth error. 2. Coarsening norm influences the choice and order of appearance of coarse equations. This has a direct 5

6 impact on the structure of coarse level matrix and performance of the smoother. 3. For coarsening and interpolation, choice of the block coefficient norm can influence the convergence of the linear solver. However, best practice was not established as some norms work better for certain cases and worse for other cases. SAMG algorithm is parallelised for both scalar and block matrices and more details on parallelisation and parallel performance will be given in the full paper. References [1] Y. Saad, Iterative methods for sparse linear systems. SIAM, [2] U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid. Elsevier, Academic Press, [3] K. Stueben, A review of algebraic multigrid, Journal of Computational and Applied Mathematics, vol. 128, pp , [4] T. Clees, AMG strategies for PDE systems with applications in industrial semiconductor simulation, PhD Thesis, University of Cologne, Germany, [5] M. Zedan and G. Schneider, A strongly implicit simultaneous variable solution procedure for velocity and pressure in fluid flow problems, in AIAA 18th Thermophysics Conference, Montreal, Canada, [6] M. Raw, A coupled algebraic multigrid method for the 3D Navier-Stokes Equations, Fast solvers for flow problems, vol. 49, pp , [7] B. Hutchinson, P. Galpin, and G. Raithby, Application of additive correction multigrid to the coupled fluid flow equations, Numerical heat transfer, vol. 13, pp , [8] S. Patankar and D. Spalding, A calculation procedure for heat, mass and momentum transfer in threedimensional parabolic flows, International journal of heat and mass transfer, vol. 15, pp ,

What is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.

What is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear. AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to