Robust iterative finite element solver for multi-terminal power distribution network resistance extraction

In reservoir simulation, algebraic multigrid (AMG) solvers are the most effective solvers for linear systems arising from the discretization of pressure equations. The AMG solvers are also important components for many multi-stage preconditioners. In this paper, we introduce our work on developing these solvers on NVIDIA Tesla GPUs. A classical algebraic multigrid solver is developed. Numerical experiments are performed to test the efficiency of this GPU-based parallel AMG solver. The numerical experiments show that the GPU-based algebraic solver is sped up to 12.5 times faster than the corresponding CPU-based algebraic multigrid solver.

We propose a new, efficient, adaptive algebraic multigrid (AMG) solver strategy for the discrete systems of partial differential equations arising from structured or unstructured grid models in reservoir simulation. The proposed strategy has been particularly tailored to linear systems of equations arising in adaptive implicit methods. The coarsening process of the AMG method designed automatically employs information on the physical structure of the models; as a smoother, an adaptive ILUT method is employed, taking care of an efficient solution of the hyperbolic parts whilst providing adequately smooth errors for the elliptic parts. To achieve a good compromise of high efficiency and robustness for a variety of problem classes – ranging from simple, small black-oil to challenging, large compositional models – an automatic, adaptive ILUT parameter and AMG solver switching strategy, α-SAMG, has been developed. Its efficiency is demonstrated for eight industrial benchmark cases by comparison against standard one-level and AMG solvers as well as the pure one- level variant of the proposed new strategy. In addition, very promising results of first parallel runs are shown.

This paper describes the development of a robust multigrid, finite element‐based, Laplace solver for accurate capacitance extraction of conductors embedded in multi‐layer dielectric domains. An algebraic multigrid based on element interpolation is adopted and streamlined for the development of the proposed solver. In particular, a new, node‐based agglomeration scheme is proposed to speed up the process of agglomeration. Several attributes of this new method are investigated through the application of the Laplace solver to the calculation of the per‐unit‐length capacitance of configurations of parallel, uniform conductors embedded in multi‐layer dielectric substrates. These two‐dimensional configurations are commonly encountered as high‐speed interconnect structures for integrated electronic circuits. The proposed method is shown to be particularly robust and accurate for structures with very thin dielectric layers characterized by large variation in their electric permittivities. More specifically, it is demonstrated that for such geometries the proposed node‐based agglomeration systematically reduces the problem size and speeds up the iterative solution of the finite element matrix. Copyright © 2007 John Wiley & Sons, Ltd.

A new high-order compact (HOC) cascadic multigrid (MG) strategy is developed for modeling large-scale complex gravity problems on nonuniform rectilinear grids. First, the governing 3D Poisson’s gravitational potential equation is discretized by a novel HOC difference scheme, which allows for Robin boundary conditions to further reduce the computational domain. Then, the resulting asymmetric linear system of equations is solved by a generalized extrapolation cascadic MG (gEXCMG) method augmented with a novel MG prolongation operator. The quartic interpolation and completed Richardson extrapolation assist this MG prolongation operator to produce a good initial guess for the biconjugate gradient stabilized smoother. Two synthetic and SEG/EAGE salt models are examined to verify the presented approach. Results indicate that our algorithm can efficiently and accurately produce the gravity signals for complex geologic models. Moreover, the comparisons with the state-of-the-art algebraic MG solvers demonstrate that our gEXCMG solver offers substantially better efficiency with significantly less memory consumption. Therefore, our newly developed method has the potential for serving as a forward engine in large-scale gravity inversions.

Solving the Poisson equation on small aspect ratio domains using unstructured meshes

Efficient forward modeling of electromagnetic (EM) fields is the basis of interpretation and inversion of the EM data, which plays an important role in practical geophysical exploration. A novel extrapolation cascadic multigrid (EXCMG) method is developed to solve large linear systems encountered in geophysical EM modeling. The original curl-curl equation with arbitrary anisotropic conductivity is regularized by including the gradient of a scaled divergence correction term, and a linear edge element method is used to discretize the equation. We develop a novel approach to address the issue of edge unknowns in 3D edge element discretizations on nonuniform rectilinear grids. Inspired by the original EXCMG method for nodal elements, we introduce a new prolongation operator that treats edge unknowns as defined on the midpoints of edges. This operator aims to provide an accurate approximation of the finite-element solution on the refined grid. Using a good initial guess significantly reduces the number of iterations required by the preconditioned biconjugate gradient stabilized solver, which is used as a smoother for the EXCMG algorithm. Numerical experiments are carried out to validate the accuracy and efficiency of our EXCMG method, including models from magnetotellurics and controlled-source EM modeling. The testing results indicate that EXCMG is more efficient than traditional Krylov-subspace iterative solvers, the algebraic multigrid solver, and those depending on the auxiliary-space Maxwell solver, especially for large-scale problems where the number of unknowns exceeds 10 million. The EXCMG method can also be applied to solve other large-scale forward modeling problems encountered in geophysics.

Iterative linear equation solvers depend on high quality preconditioners to achieve fast convergence. For sparse symmetric systems arising from large power grid analysis problems, however, preconditioners generated by traditional incomplete Cholesky factorizations are usually of low quality, resulting in slow convergence. On the other hand, although preconditioners generated by random walks are quite effective to reduce the number of iterations, it takes considerable amount of time to compute them in a stochastic manner. In this paper, we propose a new preconditioning technique for power grid analysis, named deterministic random walk, that combines the advantages of the above two approaches. Our proposed algorithm computes the preconditioners in a deterministic manner to reduce computation time, while achieving similar quality as stochastic random walk preconditioning by modifying fill-ins to compensate dropped entries. We have proved that for such compensation scheme, our algorithm will not fail for certain matrix orderings, which otherwise cannot be guaranteed by traditional incomplete factorizations. We demonstrate that by incorporating our proposed preconditioner, a conjugate gradient solver is able to outperform a state-of-the-art algebraic multigrid preconditioned solver on public IBM power grid benchmarks for DC power grid analysis, while potentially remaining very efficient for transient analysis.

In this study, jet temperature effects on afterbody drag in subsonic, transonic and supersonic flow conditions have been investigated for a wide range of nozzle throat total pressure to free-stream static pressure ratio (NPR) at Mach numbers 0.6, 0.9 and 1.2 using CFD. Preliminary CFD simulations are conducted for cold air flow with constant specific heat. Experimental data available in NASA Technical Paper 1766 is utilized for validation purposes thus, the CFD simulations are carried out for the identical geometry and boundary conditions reported in the technical paper. Depending on the Mach number, either pressure-based solver with coupled algebraic multigrid scheme or implicit density-based flow solver with Roe-FDS are used for the simulation of compressible flows to maintain convergence and obtain accurate solutions. Second order upwind schemes are preferred for all simulations and it is shown that convergence problems could be prevented when density based solvers are used at relatively high Mach numbers. For all simulations SST k-ω turbulence model proposed by Menter is selected. Based on a robust and verified CFD approach for cold jet analysis, the average drag coefficients Cd at various NPR values between 1.5 and 7 have been successfully estimated at all Ma with relative errors ranging from 5 to 15 % compared to experimental data. Then, the same numerical approach is adopted to the hot jet analysis. Further comparisons to empirical relations revealed a satisfactory agreement only at Ma = 1.2, but no acceptable match at Ma = 0.6 and 0.9.

Chapter 43 - Numerical Studies on a ccNUMA Computer Architecture for a Large Scale Race Car Aerodynamics Simulation

Sparse Matrix Vector multiplication (SpMV) is an important kernel in both traditional high performance computing and emerging data-intensive applications. By far, SpMV libraries are optimized by either application-specific or architecture-specific approaches, making the libraries become too complicated to be used extensively in real applications. In this work we develop a Sparse Matrix-vector multiplication Auto-Tuning system (SMAT) to bridge the gap between specific optimizations and general-purpose usage. SMAT provides users with a unified programming interface in compressed sparse row (CSR) format and automatically determines the optimal format and implementation for any input sparse matrix at runtime. For this purpose, SMAT leverages a learning model, which is generated in an off-line stage by a machine learning method with a training set of more than 2000 matrices from the UF sparse matrix collection, to quickly predict the best combination of the matrix feature parameters. Our experiments show that SMAT achieves impressive performance of up to 51GFLOPS in single-precision and 37GFLOPS in double-precision on mainstream x86 multi-core processors, which are both more than 3 times faster than the Intel MKL library. We also demonstrate its adaptability in an algebraic multigrid solver from Hypre library with above 20% performance improvement reported.

Laplacian matrices of graphs arise in large-scale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data, and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A lean algebraic multigrid (LAMG) solver of the symmetric linear system $Ax=b$ is presented, where $A$ is a graph Laplacian. LAMG's run time and storage are empirically demonstrated to scale linearly with the number of edges. LAMG consists of a setup phase, during which a sequence of increasingly coarser Laplacian systems is constructed, and an iterative solve phase using multigrid cycles. General graphs pose algorithmic challenges not encountered in traditional multigrid applications. LAMG combines a lean piecewise-constant interpolation, judicious node aggregation based on a new node proximity measure (the affinity), and an energy correction of coarse-level systems. This results in fast convergence and substantial setup and memory savings. A serial LAMG implementation scaled linearly for a diverse set of 3774 real-world graphs with up to 47 million edges, with no parameter tuning. LAMG was more robust than the UMFPACK direct solver and combinatorial multigrid (CMG), although CMG was faster than LAMG on average. Our methodology is extensible to eigenproblems and other graph computations.

To apply WLP to various devices from small pin count to high pin count devices, various types of WLP have been developed and evaluated in terms of package size, structure and materials. Firstly for the application of small pin count, the reliability for Wafer Level Package of 6×6 ball array (36 balls) in 0.5mm pitch had been evaluated over TC (Temperature Cycling) and HTS (High Temperature Storage) at package level and passed. Then, the reliability for WLP of 9×9 ball array (65 balls, depopulated array) in 0.4mm pitch has been evaluated by employing TC, THT (Temperature Humidity Test), 4-point bending and Drop test at board level. All test items have been passed. To verify the application of high pin count devices, Wafer Level Packages of solder ball array 12×12 (144 balls) and 14×14 (196 balls) in 0.4mm pitch have been fabricated and tested over various reliability tests. Package level reliability tests for precondition level 3, multi-reflow, PCT, HTS and TC have been evaluated and passed. Board level reliability has been proven through various reliability tests. For one package for both small and high pin devices, Wafer Level System in Package(WLSiP) which has two chips inside one WLP has been developed. To optimize the structure and materials for WLSiP, various reliability tests had been performed in terms of die size and material type. According to the above results, WLSiP test vehicle has been designed and fabricated to evaluate the package level and board level reliability tests.

We implemented a novel multi-resolution grid approach to direct current resistivity (DCR) modeling in 3-D. The multi-resolution grid was initially developed to solve the electromagnetic forward problem and helped to improve the modeling efficiency. In the DCR forward problem, the distribution of the electric potentials in the subsurface is estimated. We consider finite-difference staggered grid discretization, which requires fine grid resolution to accurately model electric potentials around the current electrodes and complex model geometries near the surface. Since the potential variations attenuate with depth, the grid resolution can be decreased correspondingly. The conventional staggered grid fixes the horizontal grid resolution that extends to all layers. This leads to over-discretization and therefore unnecessary high computational costs (time and memory). The non-conformal multi-resolution grid allows the refinement or roughening for the grid’s horizontal resolution with depth, resulting in a substantial reduction of the degrees of freedom, and subsequently, computational requirements. In our implementation, the coefficient matrix maintains its symmetry, which is beneficial for using the iterative solvers and solving the adjoint problem in inversion. Through comparison with the staggered grid, we have found that the multi-resolution grid can significantly improve the modeling efficiency without compromising the accuracy. Therefore, the multi-resolution grid allows modeling with finer horizontal resolutions at lower computational costs, which is essential for accurate representation of the complex structures. Consequently, the inversion based on our modeling approach will be more efficient and accurate.

The paper deals with a fast computational method for discretized optimal shape design problems governed by 2–dimensional magnetostatics. We discretize the underlying state problem using linear Lagrange triangular finite elements and in the optimization we eliminate the state problem for each shape design. The shape to be optimized is the interface between the ferromagnetic and air domain. The novelty of our approach is that shape perturbations do not affect grid nodal displacements, which is the case of the traditional moving–grid approach, but they are rather mapped to the coefficient function of the underlying magnetostatic operator. The advantage is that there is no additional restriction for the shape perturbations on fine discretizations. However, this approach often leads to a decay of the finite element convergence rate, which we discuss. The computational efficiency of our method relies on an algebraic multigrid solver for the state problem, which is also described in the paper. At the end we present numerical results.

Due to the advantages of joint level tests, such as low cost and flexibility, it is always desirable to predict the board level pad cratering strength by a joint level test. In order to achieve this objective, the correlation between the joint level and the board level tests must be fully understood. Nevertheless, a precise correlation between the two types of tests for pad cratering evaluation is yet to be defined. This study investigates the correlation methodology of critical failure factors between joint and board level tests on pad cratering failure mode. An intermediate critical failure factor is taken from joint level testing as a failure criterion for failure prediction in board level testing. From the results of the experimental study and finite element analysis (FEA), the critical failure factor correlation is established between the board level and the joint level test. Subsequently the obtained failure criterion is also verified by a board level experimental study. In consequence, this indirect correlation method has the capability to predict the board level pad cratering failure with various geometric parameters.