Conjugate Gradient Iterative Solver Research Articles

A Globalized and Preconditioned Newton-CG Solver for Metric-Aware Curved High-Order Mesh Optimization

We present a specific-purpose globalized and preconditioned Newton-CG solver to minimize a metric-aware curved high-order mesh distortion. The solver is specially devised to optimize curved high-order meshes for high polynomial degrees with a target metric featuring non-uniform sizing, high stretching ratios, and curved alignment — exactly the features that stiffen the optimization problem. To this end, we consider two ingredients: a specific-purpose globalization and a specific-purpose Jacobi-iLDLT(0) preconditioning with varying accuracy and curvature tolerances (dynamic forcing terms) for the CG method. These improvements are critical in stiff problems because, without them, the large number of non-linear and linear iterations makes curved optimization impractical. First, to enhance the global convergence of the non-linear solver, the globalization strategy modifies Newton’s direction to a feasible step. In particular, our specific-purpose globalization strategy memorizes the length of the feasible step (step-length continuation) between the optimization iterations while ensuring sufficient decrease and progress. Second, to compute Newton’s direction in second-order optimization problems, we consider a conjugate-gradient iterative solver with specific-purpose preconditioning and dynamic forcing terms. To account for the metric stretching and alignment, the preconditioner uses specific orderings for the mesh nodes and the degrees of freedom. We also present a preconditioner switch between Jacobi and iLDLT(0) preconditioners to control the numerical ill-conditioning of the preconditioner. In addition, the dynamic forcing terms determine the required accuracy for the Newton direction approximation. Specifically, they control the residual tolerance and enforce sufficient positive curvature for the conjugate-gradients method. Finally, to analyze the performance of our method, the results compare the specific-purpose solver with standard optimization methods. For this, we measure the matrix–vector products indicating the solver computational cost and the line-search iterations indicating the total amount of objective function evaluations. When we combine the globalization and the linear solver ingredients, we conclude that the specific-purpose Newton-CG solver reduces the total number of matrix–vector products by one order of magnitude. Moreover, the number of non-linear and line-search iterations is mainly smaller but of similar magnitude.

Stochastic nonlinear multiscale computational scheme for short fiber reinforced composites to study the influence of microstructural variability on damage propagation

In this paper, a stochastic nonlinear multiscale computational scheme is proposed to study the influence of the variability in both physical parameters of constituent materials and geometrical parameters such as fiber orientation, fiber arrangement of short fiber reinforced composites on the damage propagation. The aim is to predict many probable damage patterns in a huge number of scenarios under complex strain conditions and to discuss the influential level of variability in the physical and geometrical parameters for short fiber reinforced composites. The parameterization and numerical modelling for short fiber reinforced plastics made by injection molding are conducted. The random physical parameters are considered theoretically by the first-order perturbation based stochastic homogenization method, while sampling is used for the random geometrical parameters. When the scenario becomes huge in the stochastic nonlinear analysis, a computational scheme using sub-sampling is proposed. Moreover, the stochastic prediction of the homogenized properties of the composite material and their probable degradation are discussed. The stochastic computation of microscopic strains is analyzed though the damage propagation simulation. When the geometrical variability is considered, a definition of an average strain distribution among many samples is also presented. The efficiency of the accelerated element-by-element scaled conjugate gradient iterative solver is shown by solving many three-dimension random short fiber reinforced models with hundreds of thousands of degrees of freedom. The stochastic computational scheme provides a suitable reference to predict failure under specific reliability requirements of short fiber reinforced composites or other composite materials.

Computer modeling and simulation for undersaturated primary drive recovery mechanism

The focus of this article is to test the capability of a three-dimensional undersaturated numerical reservoir simulator developed for prediction of actual field performance cases. The simulator development employs unconditional stable Crank–Nicolson second order in time approximation discretization scheme and preconditioned conjugate gradient iterative solver for the derived fluid flow equation. Furthermore, formation volume factor as a function of cell pressure is computed at each time step until bubble-point pressure is reached. The reservoir model is then coupled with a horizontal wellbore model in the simulator by placing a well optimally at the center of its drainage volume. The numerical simulation results show that the numerical solutions improve significantly with the addition of grid blocks. Graphical visualization surface and contour map plots show rapid and continuous average reservoir pressure depletion with time for an earliest possible determination and characterization of primary drive mechanisms. The rate of pressure depletion and recovery factor estimation (1.52%) are the two main characteristics used to identify depletion drive or rock and liquid expansion drive as the best-fit prospective drive for the reservoir model. The total cumulative production of 170,000 stock-tank barrels (MSTB) recovered in 1390 days is in agreement with the field data used. Sensitivity analyses with the numerical simulator illustrated the key impact of reservoir parameters on pressure depletion with less computational expensiveness. Finally, material balance incremental (0.995) and cumulative (1.005) error checks validate the simulator robustness within an acceptable tolerance limit.

FFT-Based Methods for Solving a Rough Adhesive Contact: Description and Convergence Study

The contact problem of a semi-infinite elastic body with a rigid rough surface with adhesive forces is modeled using a boundary element method (BEM). An original theoretical framework for this problem is presented. Four original BEM algorithms are studied, for both prescribed normal pressure and prescribed penetration, in primal and dual forms. They are all based on a conjugate gradient iterative solver. For each case, the solver is finely tuned to tackle the current problem. Using fast Fourier transforms (FFT) speeds up the computation of the matrix–vector multiplications. A new way of computing the coefficients of the conjugate gradient solver which reduces the number of FFTs at each step is presented. The stability and efficiency of the different methods are compared, showing a great sensitivity to the Tabor coefficient and to the contact area ratio. The most stable algorithm proved to be reliable for very large-scale computations through a participation to M. Mueser’s Contact Mechanics Challenge.

HotSpot Thermal Floorplan Solver Using Conjugate Gradient to Speed Up

We proposed to use the conjugate gradient method to effectively solve the thermal resistance model in HotSpot thermal floorplan tool. The iterative conjugate gradient solver is suitable for traditional sparse matrix linear systems. We also defined the relative sparse matrix in the iterative thermal floorplan of Simulated Annealing framework algorithm, and the iterative method of relative sparse matrix could be applied to other iterative framework algorithms. The experimental results show that the running time of our incremental iterative conjugate gradient solver is speeded up approximately 11x compared with the LU decomposition method for case ami49, and the experiment ratio curve shows that our iterative conjugate gradient solver accelerated more with increasing number of modules.

GPU-Based Fast Decoupled Power Flow With Preconditioned Iterative Solver and Inexact Newton Method

Power flow is the most fundamental computation in power system analysis. Traditionally, the linear solution in power flow is solved by a direct method like LU decomposition on a CPU platform. However, the serial nature of the LU-based direct method is the main obstacle for parallelization and scalability. In contrast, iterative solvers, as alternatives to direct solvers, are generally more scalable with better parallelism. This study presents a fast decouple power flow (FDPF) algorithm with a graphic processing unit (GPU)-based preconditioned conjugate gradient iterative solver. In addition, the Inexact Newton method is integrated to further improve the GPU-based parallel computing performance for solving FDPF. The results show that the GPU-based FDPF maintains the same precision and convergence as the original CPU-based FDPF, while providing considerable performance improvement for several large-scale systems. The proposed GPU-based FDPF with the Inexact Newton method gives a speedup of 2.86 times for a system with over 10 000 buses if compared with traditional FDPF, both implemented based on MATLAB. This demonstrates the promising potential of the proposed FDPF computation using a preconditioned iterative solver under GPU architecture.

Toward Real-Time Finite-Element Simulation on GPU

In this paper, we introduce some parallel techniques on NVIDIA compute unified device architecture GPU for the finite-element method applied in the magnetic field computation. To ensure the load balance, each parallel thread performs the integration of one element. In the assembly step, we introduced a fast procedure based on the sorting and rearrangement of non-zero entries on the GPU global memory. Then, a reducing process is executed to obtain the resulting coefficient matrix in a sparse format. About the solving step, we use the conjugate gradient iterative solver with a variety of preconditioning techniques. Our implementation does not require any preprocessing on mesh, but takes advantage of the parallel computing power of GPU. In our test, this parallel strategy improved the performance 30 times faster on the assembly process and four times faster on the solving process.

Low-memory iterative density fitting.

A new low-memory modification of the density fitting approximation based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the number of iterations needed for convergence by up to one order of magnitude. The matrix-vector products needed within the iterative algorithm are calculated using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the standard density fitting implementation, up to 15-fold reduction of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing density functional theory calculations for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12-core CPU workstation.

On a Wavenumber Optimized Streamline Upwinding Method for Solving Steady Incompressible Navier-Stokes Equations

A stabilized upwind finite-element model is developed to solve the three-dimensional incompressible steady Navier-Stokes equations. The test function is constructed to have a larger weight on the upstream side. This has been achieved by adding a stabilized term to the shape function so as to optimize the numerical wavenumber for convection terms. To avoid Lanczos or pivoting breakdown while solving the resulting unsymmetric and indefinite mixed finite-element matrix equations iteratively, the finite-element equation has been modified by pre-multiplying it with its transpose. The resulting normalized matrix equation becomes symmetric and positive-definite. We can therefore apply a computationally efficient conjugate gradient Krylov iterative solver to get an unconditionally convergent solution. However, the condition number of the new system becomes the square of the original unsymmetric indefinite system. To fully exploit excellent convergence nature of the conjugate gradient iterative solver, an element-by-element strategy is adopted to avoid assembling of all the stiffness matrices obtained at element level. We alleviate the drawback of slower convergence of the conjugate gradient method due to the increased condition number by preconditioning the positive-definite matrix. The resulting preconditioned matrix equation is solved in a matrix-free manner using the preconditioned conjugate gradient iterative solver. The developed finite-element code is first verified by solving a problem amenable to analytical solution. The benchmark lid-driven cavity problem is also solved in a cube for assessing the three chosen iterative solvers.

Error control schemes for adaptive time integration of magnetodynamic systems with variable spatial mesh resolution

PurposeA combination of both time step adaptivity and spatial mesh adaptivity is presented for transient magneto‐quasistatic fields.Design/methodology/approachError controlled time step adaptivity is achieved using an implicit integration scheme and the spatial mesh resolution is adapted in each time step in order to effectively resolve the appearing and disappearing local transient saturation effects and eddy current layers. Two spatial refinement strategies are considered, the red‐green refinement leading to a regular mesh and the red refinement leading to an irregular mesh. Numerical results for 2d nonlinear magneto‐dynamic problems are presented.FindingsAn algorithm is proposed which computes the solution of a transient magnetostatic problem given a user prescribed error tolerance for the time stepping and the spatial refinement. The red refinement leading to irregular meshes requires projection techniques in the iterative conjugate gradient solver. However, the algorithm with red‐green refinement turns out to perform faster since the projection is too expensive.Originality/valueThe combination of error controlled time stepping and spatial adaptivity is firstly established in electromagnetic field computation.

Global Matrix-Free Finite-Element Scheme for Natural Convection in a Square Cavity with Step Blockage

A global matrix-free finite-element scheme is proposed for the solution of two-dimensional Navier-Stokes equations in velocity–vorticity form. By including the boundary conditions of the field variables at the element level itself, the global assembly of matrices is completely eliminated, thus resulting in a significant saving in computer memory. Only the global vectors obtained as a result of matrix–vector products are assembled at the time of solution of the simultaneous equations, using a conjugate gradient iterative solver. The method is validated by solving a benchmark problem on natural convection in a square cavity. The incompressibility constraint of the flow field is satisfied by imposing the vorticity definition at the boundaries using a second-order-accurate Taylor series expansion scheme. Results obtained for natural convection in a square cavity for Rayleigh number 103 < Ra < 106 indicate excellent agreement with benchmark solutions. The capability of the method to treat complex flow geometries is demonstrated by extending the study to natural convection in a square cavity with an internal square step block positioned on the left wall. Comparison of the memory storage with the compact vector storage scheme is also discussed.

Start vector generation for implicit Newmark time integration of the wave equation

Different extrapolation strategies are presented for the construction of starting values in implicit Newmark time-stepping schemes used for the solution of the electromagnetic wave equation. The implicit character of this schemes implies the solution of a system of linear equations in each time step. The choice of a good start value derived from the extrapolation of the equation system is shown to improve the convergence of the iterative conjugate gradient solver which reduces the total simulation time

Optimizing the discrete-dipole approximation for sequences of scatterers with identical shapes but differing sizes or refractive indices

We study scattering of light by small particles with identical shapes but either moderately differing sizes or refractive indices by utilizing the discrete-dipole approximation (DDA). Assuming that accurate DDA solutions are available for either a sequence of sizes or refractive indices, we initialize the iterative conjugate gradient solver for a new size or refractive index by making “educated guesses” of the electric field vectors using classical Lagrange, rational-function, and modified Adams–Bashforth–Moulton extrapolation schemes. In the present pilot study, we assess the initialization schemes for spherical and cubic particles. As compared to the common initialization using the incident electric field, we show that careful extrapolation can significantly reduce the number of iterations. At best, the computing time can decrease by an order of magnitude whereas, typically, the improvement is some tens of percent for sizes comparable to the wavelength. In solving large numbers of single-particle scattering problems, initialization via extrapolation can yield substantial savings in computing time. In particular, the present approach should prove useful when the precise scatterer sizes and refractive indices are unknown, e.g., when interpreting astronomical observations of atmosphereless solar-system objects and experimental measurements.

An efficient global matrix free finite element algorithm for 3D flow problems

This paper describes a finite element solution algorithm for the numerical solution of large size three-dimensional flow problems on a personal computer. To demonstrate the algorithm, the Stokes equations in velocity–vorticity form are solved for a lid-driven cubical cavity problem. The Galerkin's weighted residual form of the governing equations is evaluated for all the elements of the computational domain and kept as element-matrices and element-vectors. This results in a set of simultaneous equations corresponding to the global nodes of each element. Those elements that contain the boundary nodes are modified to incorporate the Dirichlet boundary conditions. A conjugate gradient iterative scheme is employed to solve the simultaneous equations in element form to get the solution at the global nodes. The matrix–vector products used in the conjugate gradient iterative solver are performed in element level, assembling only the element-level vectors to form the global vectors. Since the element-level computation has eluded the formation of global matrices, the numerical solution of three-dimensional Stokes equations using a mesh of size as high as 513 could be achieved on a personal computer. The algorithm is validated by comparing the results for a three-dimensional transient diffusion problem and Stokes flow in a lid-driven cubical cavity. Copyright © 2004 John Wiley & Sons, Ltd.

A numerical confirmation of the dual body vortex flowmeter design

In this study we demonstrate the effectiveness of adding a second bluff body as a chaos control. Numerical method involves operator splitting technique, balance tensor diffusivity (BTD), and an element-by-element conjugate gradient iterative solver. To avoid numerical instability and to satisfy the physical constrain for obtaining the lowest frequency of vortex shedding, the time step needs to be carefully selected. The simulated result shows that a second bluff body placed in the flow path can enhance the repeatability of vortex pattern and, thus, can confirm the experimental finding of Bently et al. and Okajima.

Stress–Strain Analysis of Thin-Walled Shells with Massive Ribs

A multifrontal method is proposed for solving large-scale finite-element problems with a symmetric stiffness matrix. This approach is based on node ordering rather than element ordering as in the traditional frontal or multifrontal methods. This allows us to employ the efficient well-known ordering techniques such as the minimum-degree algorithm or the nested dissection method. A thin-walled cylindrical shell with massive ribs is considered as an example. The efficiency of the method proposed is illustrated against the classical skyline and aggregation multilevel conjugate-gradient iterative solvers

A numerical confirmation of the dual body vortex flowmeter design

In this study we demonstrate the effectiveness of adding a second bluff body as a chaos control. Numerical method involves operator splitting technique, balance tensor diffusivity (BTD), and an element-by-element conjugate gradient iterative solver. To avoid numerical instability and to satisfy the physical constrain for obtaining the lowest frequency of vortex shedding, the time step needs to be carefully selected. The simulated result shows that a second bluff body placed in the flow path can enhance the repeatability of vortex pattern and, thus, can confirm the experimental finding of Bently et al. and Okajima.

Finite element analysis of vortex shedding oscillations from cylinders in the straight channel

In this paper a GCL (Geometry Conservation Law) – preserving finite element model is developed to simulate the fluid-structure interaction phenomena. The boundary locations are part of the solution procedures. Within the moving grid framework, the employed numerical method involves the operator splitting technique, balance tensor diffusivity (BTD), Runge-Kutta time-stepping method, and an element-by-element conjugate gradient iterative solver. Flows around a self-vibrating cylinder and elastically supported cylinders are investigated to validate the present method. The simulated results agree well with available data in the literature. The lock-in phenomenon that may cause the unstable motion of cylinders is also revealed. It is significant from the simulated 2×2 four cylinders result that the rear two cylinders approach to each other at a time earlier than the front two ones.

Parallel Computing Applied to Breeding Value Estimation in Dairy Cattle

Continuous genetic evaluation of dairy cattle with test-day models is desired in Finland. However, the computing time for the genetic evaluation is 4 d and exceeds the minimum of a weekend. Three parallel implementations of the preconditioned conjugate gradient iterative solver were programmed and compared to identify the best strategy for solving mixed model equations using parallel computing. The programs were used to solve two random regression test-day models with approximately 7.28 and 49.9million unknowns. The latter model will be used in the Finnish dairy cattle evaluation. Computing times for the smaller model with the four processors available were 52, 32, and 27% of the single processor program when the complexity of the parallel program was increased. In practice, the best program required the most programming because the other parallel programs could not solve the larger model because of excess memory requirements. Parallel computing with four processors reduced the time to obtain solutions of Finnish dairy cattle evaluations to under 2 d. Benefit from parallel computing will be increased if amount of computing memory is increased.

A parallel preconditioned conjugate gradient solution method for finite element problems with coarse–fine mesh formulation

The object of this paper is a parallel preconditioned conjugate gradient iterative solver for finite element problems with coarse-mesh/fine-mesh formulation. An efficient preconditioner is easily derived from the multigrid stiffness matrix. The method has been implemented, for the sake of comparison, both on a IBM-RISC590 and on a Quadrics-QH1, a massive parallel SIMD machine with 128 processors. Examples of solutions of simple linear elastic problems on rectangular grids are presented and convergence and parallel performance are discussed.