A parallel Heap-Cell Method for Eikonal equations

Numerous applications of Eikonal equations prompted the development of many efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial two-scale technique that has been shown to have advantages over other serial state-of-the-art solvers for a wide range of problems. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in $R^3$ show that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver. We further explore the influence on performance and scaling of data precision, early termination criteria, and the hardware architecture. A shorter version of this manuscript (omitting these more detailed tests) has been submitted to SIAM Journal on Scientific Computing in 2012.Comment: (a minor update to address the reviewers' comments) 31 pages; 15 figures; this is an expanded version of a paper accepted by SIAM Journal on Scientific Computin

A fast GPU Monte Carlo Radiative Heat Transfer Implementation for Coupling with Direct Numerical Simulation

We implemented a fast Reciprocal Monte Carlo algorithm, to accurately solve radiative heat transfer in turbulent flows of non-grey participating media that can be coupled to fully resolved turbulent flows, namely to Direct Numerical Simulation (DNS). The spectrally varying absorption coefficient is treated in a narrow-band fashion with a correlated-k distribution. The implementation is verified with analytical solutions and validated with results from literature and line-by-line Monte Carlo computations. The method is implemented on GPU with a thorough attention to memory transfer and computational efficiency. The bottlenecks that dominate the computational expenses are addressed and several techniques are proposed to optimize the GPU execution. By implementing the proposed algorithmic accelerations, a speed-up of up to 3 orders of magnitude can be achieved, while maintaining the same accuracy

Hierarchical Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Simulations

In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body $p$ consisting of $n$ beads, the $6 \times 3n$ transformation matrix $Z$ that maps the force on each bead to $p$'s translational and rotational forces (a $6\times 1$ vector), and $V$ the row space of $Z$, we show how to explicitly construct the $(3n-6) \times 3n$ matrix $\tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{\perp}$ (orthogonal complement of $V$) using only $\mathcal{O}(n \log n)$ operations and storage. For applications where only the matrix-vector multiplications $\tilde{Q}{\bf v}$ and $\tilde{Q}^T {\bf v}$ are needed, we introduce asymptotically optimal $\mathcal{O}(n)$ hierarchical algorithms without explicitly forming $\tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms

Domain decomposition methods for the parallel computation of reacting flows

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors

The block adaptive multigrid method applied to the solution of the Euler equations

In the present study, a scheme capable of solving very fast and robust complex nonlinear systems of equations is presented. The Block Adaptive Multigrid (BAM) solution method offers multigrid acceleration and adaptive grid refinement based on the prediction of the solution error. The proposed solution method was used with an implicit upwind Euler solver for the solution of complex transonic flows around airfoils. Very fast results were obtained (18-fold acceleration of the solution) using one fourth of the volumes of a global grid with the same solution accuracy for two test cases

Wall Distance Evaluation Via Eikonal Solver for RANS Applications

RÉSUMÉ Les logiciels de mécanique des fluides assistée par ordinateur (CFD) sont de plus en plus utilisés pour la conception d’aéronefs. L’utilisation de grappes informatiques haute performance permet d’augmenter la puissance de calcul, aux prix de modifier la structure du code. Dans les codes CFD, les équations de Navier-Stokes moyennées (plus connues sous le nom des équations RANS) sont souvent résolues. Par conséquent, les modèles de turbulence sont utilisés pour approximer les effets de la turbulence. Dans l’industrie aéronautique, le modèle Spalart-Allmaras est bien accepté. La distance à la paroi dans ce modèle, par exemple, joue un rôle clé dans l’évaluation des forces aérodynamiques. L’évaluation de ce paramètre géométrique doit alors être précis et son calcul efficace. Avec les nouveaux développement des hardwares, un besoin se crée dans la communauté afin d’adapter les codes CFD à ceux-ci. Les algorithmes de recherche comme les distances euclidienne et projetée sont des méthodes souvent utilisées pour le calcul de la distance à la paroi et ont tendance à présenter une mauvaise scalabilité. Pour cette raison, un nouveau solveur pour la distance à la paroi doit être développé. Pour utiliser les solveurs et techniques d’accélération déjà existantes au sein du code CFD, l’équation Eikonal, une équation aux différentielles partielles non-linéaires, a été choisie. Dans la première partie du projet, le solveur d’équation Eikonal est développé en 2D et est résolue dans sa forme advective au centre de cellule. Les méthodes des différences finies et des volumes finis sont testées. L’équation est résolue à l’aide d’une discrétisation spatiale de premier ordre en amont. Les solveurs ont été vérifiés sur des cas canoniques, tels une plaque plane et un cylindre. Les deux méthodes de discrétisation réussissent à corriger les effets de maillages obliques et courbes. La méthode des différences finies possède un taux de convergence en maillage de deuxième ordre tandis que la méthode des volumes finis a un taux de convergence de premier ordre. L’addition d’une reconstruction linéaire de la solution à la face permet d’étendre la méthode des volumes finis à une méthode de deuxième ordre. De plus, les méthodes de différence finie et de volume fini de deuxième ordre permettent de bien représenter la distance à la paroi dans les zones de fort élargissement des cellules. L’équation Eikonal est ensuite vérifié sur plusieurs cas dont un profil NACA0012 en utilisant trois modèles de turbulence : Spalart-Allmaras, Menter SST et Mener-Langtry SST transitionnel.----------ABSTRACT Computational fluid dynamics (CFD) software is being used more often nowadays in aircraft design. The use of high performance computing clusters can increase computing power, but requires change in the structure of the software. In the aeronautical industry, CFD codes are often used to solve the Reynolds-Averaged Navier-Stokes (RANS) equations, and turbulence models are frequently used to approximate turbulent effects on flow. The Spalart-Allmaras turbulence model is widely accepted in the industry. In this model, wall distance plays a key role in the evaluation of aerodynamic forces. Therefore calculation of this geometric parameter needs to be accurate and efficient. With new developments in computing hardware, there is a need to adapt CFD codes. Search algorithms such as Euclidean and projected distance are often the methods used for computation of wall distance but tend to exhibit poor scalability. For this reason, a new wall distance solver is developed here using the Eikonal equation, a non-linear partial differential equation, chosen to make use of existing solvers and acceleration techniques in RANS solvers. In the first part of the project, the Eikonal equation solver was developed in 2D and solved in its advective form at the cell center. Both finite difference and finite volume methods were tested. The Eikonal equation was also solved using a first-order upwind spatial discretization. The solvers were verified through canonical cases like a flat plate and a cylinder. Both methods were able to correct the effects of skewed and curved meshes. The finite difference method converged at a second-order rate in space while the finite volume method converged at a first-order rate. The addition of a linear reconstruction of the solution at the face extended the finite volume method to a second-order method. Moreover, both finite difference and second-order finite volume methods were well represented by wall distance in zones of strong cell growth. The finite difference method was chosen, as it required less computing time. The Eikonal equation was then verified for several cases including a NACA0012 using three turbulence models: Spalart-Allmaras, Menter’s SST and Menter-Langtry transitional SST. For the first model, the Eikonal equation was able to correct grid skewness on the turbulent viscosity as well as on the aerodynamic coefficients, while for the other two yielded results similar to Euclidean and projected distance. To verify the implementation and convergence of the multi-grid scheme, the new wall distance solver was tested on an ice-accreted airfoil. In addition, the overset grid capabilities of the wall distance solver were verified on the McDonnell Douglas airfoil. Finally, the DLR-F6, a 3D case, was solved to show that the Eikonal equation can be extended to 3D meshes