Zdenek Johan and Thomas J. R. Hughes. A globally convergent matrixfree algorithm for implicit time-marching schemes arising in nite element analysis in uids. Computer methods in applied mechanics and engineering, 87:281-304, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....format yielding successive linear problems. This time integrator is described in detail in Jansen et al. 22] Subsequently, each linear problem is solved using the Matrix Free Generalized Minimal RESidual (MF GMRES) solution technique with a block diagonal preconditioner developed by Johan et a1. [23] or Element by Element GMRES techniques. Convergence of the non linear problem is confirmed before moving to the next time step. The nature of these finite element computations, where most all work is in computing local element integrals, lends itself extremely well to use on parallel computers. ....

Z. Johan, T. J. R. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time marching schemes arising in finite element analysis. Comp. Meth. Appl. Mech. Engng., 87:281-304, 1991.

....nite element solution of the nonlinear Navier Stokes equations. The nonlinear part is tackled with a variant of the Newton Krylov method. We omit the description of the uid dynamics equations, discretization and numerical schemes, that are out of the scope of this study, and refer the reader to [106, 107, 157] for a complete description of these computational uid dynamics aspects. We rather focus on the description of the preconditioners to solve the linear systems involved at each step of the nonlinear iterations. This study was conducted in the framework of an industrial collaboration with ....

J. Zdenek, T. J.R. Hughes, and S. Farzin. A globally convergent matrix-free algorithm for implicit time marching schemes arising in nite element analysis in uids. Comp. Meth. in

....method. If the ILU preconditioner is employed, a lower order approximation to the Jacobian matrix can be used in its calculation in order to conserve memory resources. Full Newton algorithms using either full matrix or matrix free implementations of Krylov methods are described in references [2, 4, 10, 11, 14, 16]. The storage requirement can also be reduced by using a quasi Newton method in which the exact Jacobian matrix is replaced by a lower order approximation [1, 3, 5, 7, 12, 17, 21] Although the number of Newton iterations to convergence may increase, the cost per GMRES iteration is reduced due to ....

Johan, Z., T. J. R. Hughes and F. Shakib, "A globally convergent matrix-free algorithm for implicit timemarching schemes arising in finite element analysis in fluids," Computer Methods in Applied Mechanics and Engineering, vol. 87, pp. 281--304, 1991.

....systems iteratively using the GMRES algorithm [19] For very large systems of equations, we use a matrix free iteration strategy to obtain the solution of the nonlinear system. This element vectorbased computation totally eliminated the need to form any matrices, even at the element level [4,20]. 6 Parallel Implementation The computation of 3D free surface flow applications are of very large scale. Parallel supercomputers with hundreds of fast processors, such as CRAY T3E and IBM SP, are used to reduce the computational time. In the parallel implementation, we use a messagepassing ....

Z. Johan, T.J.R. Hughes and F. Shakib, "A Globally Convergent Matrix-free Algorithm for Implicit Time-marching Schemes Arising in Finite Element Analysis in Fluids," Computer Methods in Applied Mechanics and Engineering, 87 (1991), 281-304. 10

.... [18] and Brown Hindmarsh [9] and the PDE oriented work of Brown Saad [10] The term Newton Krylov seems first to have been applied to such problems in [10] The GMRES [67] method was firmly established in CFD following the work of Wigton, Yu, Young [91] and Johann, Hughes, Shakib [38, 70]. Venkatakrishnan Mavriplis showed in [86] that NK methods (preconditioned with a global incomplete factorization) are competitive with multigrid methods for large scale CFD problems; a similar comparison for the matrix free form of such methods was given by Keyes in [43] A study of the ....

Z. Johann, T. J. R. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Computational Methods in Applied Mechanics and Engineering, 87:281--304, 1991.

.... [18] and Brown Hindmarsh [9] and the PDE oriented work of Brown Saad [10] The term Newton Krylov seems first to have been applied to such problems in [10] The GMRES [67] method was firmly established in CFD following the work of Wigton, Yu, Young [91] and Johann, Hughes, Shakib [38, 70]. Venkatakrishnan Mavriplis showed in [86] that NK methods (preconditioned with a global incomplete factorization) are competitive with multigrid methods for large scale CFD problems; a similar comparison for the matrix free form of such methods was given by Keyes in [43] A study of the ....

Z. Johann, T. J. R. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Computational Methods in Applied Mechanics and Engineering, 87:281--304, 1991.

....for explicit methods. Thus, for structured meshes, work has been done both on the development of new implicit schemes [LS88] Sen90] and on the use of new tools to solve large sparse non symmetric linear systems[WYY85] HTS93] Also for unstructured meshes, new schemes[HFM86] SF94] and techniques [JHF91], Dut91] LF94] Lan94] have been developped. However, in most cases, each implicit step leads to one costly non linear system to be solved. When three dimensional space problems and or chemical context are considered, the size of the systems is usually large. Thus, the memory size of current ....

Z. Johan, T. Hughes, and Shakib F. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Comp. Meth. in Apllied Mech. and. Eng., 87:281--304, 1991.

....is freedom from explicit computation of the elements of the Jacobian matrix, through directional differencing. Early references to so called matrix free methods from the ODE literature may be found in [3, 8] Important introductions into the computational aerodynamics community occurred in [22, 34] and a general PDE treatment was given in [4] Instead of the full Newton correction, inexact Newton methods work only to satisfy jjJ(u l )ffiu l f(u l )jj 2 l jjf(u l )jj 2 ; for some sequence l 0. It may be shown [11] that, for iterates in the domain of convergence of Newton s ....

Z. Johann, T. J. R. Hughes and F. Shakib (1991): A Globally Convergent Matrix-Free Algorithm for Implicit Time-Marching Schemes Arising in Finite Element Analysis in Fluids, Comp. Meths. Appl. Mech. Engrg. 87 281--304.

....readily available when integrating an ODE system. For example, the solution at the last timestep, or the solution derived from a less accurate solution method, such as an explicit or multistep solver, are both good possibilities. We have implemented the parallel ODE solver with matrix free methods [18] [4] where matrices are not assembled as collections of numbers, but are instead passed around as functions. The kernel of matrix free methods may be stated quite simply: The fundamental concept is a linear transformation, not the matrix that represents it. Iterative methods access the matrix ....

Johan, Z., Hughes, T.J., and Shakib, F., A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Comput. Meth. Appl. Mech. Engr. 87 (1991), 281-304.

....these systems with the Newton Raphson method. At each Newton Raphson step, we need to solve a linear equation system for the increment vector. These linear systems are solved iteratively, where at each iteration the residual of the system is formed by an elementvector based (matrix free) method [13]. The other components of this iterative strategy are a diagonal preconditioner and the GMRES update technique [14] The methods described above have been implemented on parallel computing systems: a 512 node Thinking Machines CM 5, a 512 node CRAY T3D, and a 20 processor SGI Onyx [15 18] In ....

Z. Johan, T.J.R. Hughes, and F. Shakib, "A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids", Computer Methods in Applied Mechanics and Engineering, 87 (1991) 281--304.

....for iterative solution methods. Matrix free approaches, in general, do not require the explicit assembly of any global matrices and therefore result in substantial storage reductions. Such approaches have also been used in the context of large scale problems arising in computational fluid dynamics [5]. The matrix free interpretation of DtN described here allows the use of this exact boundary condition without any storage penalties related to its non local nature, thereby rendering it as efficient as local boundary conditions. 2 Finite Element Formulation Consider the reduced wave equation in ....

Z. Johan, T. J. R. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Comp. Methods in Applied Mech. Engrg., 87:281--304, 1991.

....for explicit methods. Thus, for structured meshes, work has been done both on the development of new implicit schemes [LS88] Sen90] and on the use of new tools to solve large sparse non symmetric linear systems[WYY85] HTS93] Also for unstructured meshes, new schemes[HFM86] SF94] and techniques [JHF91], Dut91] LF94] Lan94] have been developped. However, in most cases, each implicit step leads to one costly non linear system to be solved. When three dimensional space problems and or chemical context are considered, the size of the systems is usually large. Thus, the memory size of current ....

Z. Johan, T. Hughes, and Shakib F. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Comp. Meth. in Apllied Mech. and. Eng., 87:281--304, 1991.

....are good estimates of those obtained in practice. 2. Potential cost savings in the operations and storage required for the Au product can be realized by employing a matrix free implementation, i.e. by performing the multiplication prior to the integration which leads to the formation of A (see [86, 110] and references therein) This procedure is particularly advantageous in cases that require several matrix multiplications to form A, such as the regularized BurtonMiller formulation [6] The drawback of this approach for comparison purposes is that the effects of solving the equations become ....

....splitting. It is also noteworthy that the preconditioners MHB and MHBDS are not based on a splitting or factorization of the coecient matrix and hence do not explicitly require the matrix AN. Iterative solution strategies on massively parallel computers often employ matrix free approaches [86 110 111] to fully exploit the high degree of data parallelism intrinsic in finite element computations. In such cases, the coecient matrix A is never explicitly formed and due to this usual preconditioners entail significant storage and computational overheads. However, matrix free iterative strategies ....

Z. Johan, T. J. R. Hughes, and F. Shakib. A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids. Computer Methods in Applied Mechanics and Engineering, 87:281-304, 1991.

....of e v at iterations i and i 1, respectively. e R is the residual of the nonlinear problem and e J is the consistent Jacobian associated with e R. The consistent Jacobian is often replaced by a Jacobian like matrix e J leading to a more stable time marching algorithm (see Johan, et al. [14]) A residual like vector e R associated with e J can be defined as e J def = e R e v (6) Z.Johan, T.J.R.Hughes, K.K.Mathur and S.L.Johnsson preprint 7 The complete algorithm is summarized in Box 1. In this algorithm, Nmax is the maximum number of time steps; i max is the maximum number ....

....R (10) This preconditioned system of equations is solved using the Generalized Minimal RESidual (GMRES) algorithm. This algorithm was introduced by Saad and Schultz [16] Its effectiveness for computational fluid dynamics problems has been demonstrated by several research groups (see, for example, [14, 15, 17, 18]) The GMRES algorithm computes an approximate solution p 0 z, where p 0 is an initial guess (usually taken to be 0) and z is in the Krylov space K def = fr 0 ; J r 0 ; J k Gamma1 r 0 g. r 0 = GammaR Gamma J p 0 is the residual and k is the dimension of K. The vector z is solution ....

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Z. Johan, T.J.R. Hughes and F. Shakib, "A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids," Computer Methods in Applied Mechanics and Engineering, 87 (1991) 281--304.

....as conjugate gradient or GMRES, the solution update phase requires daxpy and dot product operations, as well as matrix vector products. The matrix vector product is performed either in an elementby element fashion using element left hand side matrices [17, 25] or using a matrix free 2 strategy [12]. In both cases, the matrix vector product is done through a gather computescatter cycle similar to Steps 4 through 6 in the above pseudo code. We can therefore describe a finite element program as being a loop over a certain number of iterations (iterations being either time steps or a ....

....An implicit time marching scheme is used to solve the nonlinear problems arising from the finite element discretization. Each nonlinear system is linearized using a Newton type method and each nonsymmetric system of equations is solved using a matrix free preconditioned GMRES algorithm [12, 24, 25]. All examples presented in this section involve the computation of steady inviscid flows. We have therefore used a local time stepping strategy where the time step in each element is a function of a global CFL number. One integration point per element was used. Only one Newton iteration per time ....

Z. Johan, T.J.R. Hughes and F. Shakib, "A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids," Computer Methods in Applied Mechanics and Engineering, 87 (1991) 281--304.

....below have been solved using a data parallel finite element program written in CM Fortran. The variational form is based on the Galerkin least squares formulation [10, 11, 12, 13] A matrix free implicit iterative solver based on the GMRES algorithm is used to converge solutions to steady state [16, 22, 23]. The reader should refer to [14] and [15] for details related to the data parallel implementation. The mesh partitioning algorithm and the communication primitives presented above have been used in the current version of the program. The tolerance and the Krylov space size in GMRES are set to 0.1 ....

Z. Johan, T.J.R. Hughes and F. Shakib, "A globally convergent matrix-free algorithm for implicit time-marching schemes arising in finite element analysis in fluids," Computer Methods in Applied Mechanics and Engineering, 87 (1991) 281--304.

No context found.

Zdenek Johan and Thomas J. R. Hughes. A globally convergent matrixfree algorithm for implicit time-marching schemes arising in nite element analysis in uids. Computer methods in applied mechanics and engineering, 87:281-304, 1991.

No context found.

Z. Johan, T.J.R. Hughes, and F. Shakib. "A Globally Convergent Matrix-free Algorithm for Implicit Time-marching Schemes Arising in Finite Element Analysis in Fluids". Comp. Meth. in App. Mec. and Eng., 87, 1991.

No context found.

Johan, Z., Hughes, T.J.R., and Shakib, F., "A Globally Convergent Matrix-Free Algorithm for Implicit Time-Marching Schemes Arising in Finite Element Analysis in Fuids," Comput. Methods Appl. Mech. Engrg., vol. 87, pp. 281--304, 1991.

No context found.

Johan Z., Hughes T. J. R. and Shakib, F., 1991. "A globally convergent matrix--free algorithm for implicit time--marching schemes arising in finite element analysis in fluids", Comp. Meth. Appl. Mech. and Engr., Vol. 87, 281--304.

No context found.

T.J.R. Hughes Z. Johan and F. Shakib. A globally convergent matrix-free algorithm for implicit time marching schemes arising in finite element analisys of fluids. Comput. Methods Appl. Mech. Engrg., 87, 281--304 (1991).

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