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• Deflated preconditioned conjugate gradient solvers for linear elasticity

  R. Aubry, F. Mut, S. Dey, R. Lohner

  Int. J. Numer. Meth. Engng. (2011). Vol. 88 (11), pp. 1112-1127

  
  

  Abstract

  Extensions of deflation techniques previously developed for the Poisson equation to static elasticity are presented. Compared to the (scalar) Poisson equation (J. Comput. Phys. 2008; 227 (24):10196–10208; Int. J. Numer. Meth. Engng 2010; DOI: 10.1002/nme.2932; Int. J. Numer. Meth. Biomed. Engng 2010; 26 (1):73–85), the elasticity equations represent a system of equations, giving rise to more complex low‐frequency modes (Multigrid . Elsevier: Amsterdam, 2000). In particular, the straightforward extension from the scalar case does not provide generally satisfactory convergence. However, a simple modification allows to recover the remarkable acceleration in convergence and CPU time reached in the scalar case. Numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in CPU time for grids with moderate graph depths compared to the non‐deflated version. Furthermore, a monotonic decrease of iterations with increasing subdomains, as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included.

  Abstract
  Extensions of deflation techniques previously developed for the Poisson equation to static elasticity are presented. Compared to the (scalar) Poisson equation (J. Comput. [...]

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• Combinatorial Aspects/Algorithms in Computational Fluid Dynamics

  R. Lohner

  (2012). Combinatorial Scientific Computing, Edited ByUwe Naumann, Olaf Schenk, Chapter 9, 24 p.

  
  

  Abstract

  Most of the equations used to describe the behaviour of continua are of the form: u,t +∇ · (Fa − Fv) = S(u) , (9.1) where u,Fa,Fv,S(u) denote the vector of unknowns, advective and diffusive flux tensors, and source-terms respectively. In the case of compressible gases, we have u = (ρ ; ρvi ; ρe) , Faj = (ρvj ; ρvivj + pδij ; vj(ρe+ p)) , Fvj = (0 ; σij ; vlσlj + kT,j) . (9.2) Here ρ, p, e, T, k, vi denote the density, pressure, specific total energy, temperature, conductivity and fluid velocity in direction xi respectively. This set of equations is closed by providing an equation of state, e.g., for a polytropic gas: p = (γ − 1)ρ[e− 1 2 vjvj ] , T = cv[e − 1 2 vjvj ] , (9.3 a, b) where γ, cv are the ratio of specific heats and the specific heat at constant volume respectively. Furthermore, the relationship between the stress tensor σij and the deformation rate must be supplied. For water and almost all gases, Newton’s hypothesis σij = µ( ∂vi ∂xj + ∂vj ∂xi ) + λ ∂vk ∂xk δij (9.4) complemented with Stokes’ hypothesis λ = −2µ 3 (9.5)   is an excellent approximation. The compressible Euler equations are obtained by neglecting the viscous fluxes, i.e., setting Fv = 0. The incompressible Euler or Navier-Stokes equations are obtained by assuming that the density is constant and that pressure does not depend on temperature. The Maxwell equations of electromagnetics, the heat conduction equations of solids, and the equations describing elastic solids undergoing small deformation can readily be written in the form given by Equation (9.1).

  Abstract
  Most of the equations used to describe the behaviour of continua are of the form: u,t +∇ · (Fa − Fv) = S(u) , (9.1) where u,Fa,Fv,S(u) denote the vector [...]

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• The computation of the eddy along the upper wall in the three‐dimensional flow over a backward‐facing step

  N. Malamataris, R. Lohner

  Int. J. Numer. Meth. Fluids (2011). Vol. 68 (9), pp. 1102-1125

  
  

  Abstract

  A three‐dimensional laminar flow over a backward‐facing step is studied as a numerical experiment by solving the steady‐state, isothermal and incompressible Navier–Stokes equations using two different finite element codes. The Reynolds number ranges from 100 to 1050. The expansion ratio is 1:1.94, and the aspect ratio is 1:36.7. The numerical experiment reveals both eddies along the lower and upper walls downstream of the step. Results of computations regarding positions of detachment of the eddy along the upper wall and positions of reattachments of the eddies along both the lower and upper walls are tabulated along with positions and magnitudes of global extrema of shear rate within the eddies. The wall effects are shown by calculating streamlines along planes parallel/normal to the lateral walls of the domain and depicting how the streamlines are distorted close to the walls and how they assume a two‐dimensional configuration in the plane of symmetry. Comparisons are made with available numerical results and laboratory measurements.

  Abstract
  A three‐dimensional laminar flow over a backward‐facing step is studied as a numerical experiment by solving the steady‐state, isothermal and incompressible Navier–Stokes [...]

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• Generating seamless surfaces for transport and dispersion modeling in GIS

  F. Camelli, J. Lien, D. Shen, D. Wong, M. Rice, R. Lohner, C. Yang

  GeoInformatica (2012). Vol. 16, pp. 307-327

  
  

  Abstract

  A standard use of triangulation in GIS is to model terrain surface using TIN. In many simulation models of physical phenomena, triangulation is often used to depict the entire spatial domain, which may include buildings, landmarks and other surface objects in addition to the terrain surface. Creating a seamless surface of complex building structures together with the terrain is challenging and existing approaches are laborious, time-consuming and error-prone. We propose an efficient and robust procedure using computational geometry techniques to derive triangulated building surfaces from 2D polygon data with a height attribute. We also propose a new method to merge the resultant building surfaces with the triangulated terrain surface to produce a seamless surface for the entire study area. Using Oklahoma City data, we demonstrate the proposed method. The resultant surface is used as the input data for a simulated transport and dispersion event in Oklahoma City. The proposed method can produce the seamless surface data to be used for various types of physical models in a fraction of the time required by previous methods.

  Abstract
  A standard use of triangulation in GIS is to model terrain surface using TIN. In many simulation models of physical phenomena, triangulation is often used to depict the entire [...]

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• Semi‐automatic porting of a large‐scale Fortran CFD code to GPUs

  A. Corrigan, F. Camelli, R. Lohner, F. Mut

  Int. J. Numer. Meth. Fluids (2011). Vol. 69 (2), pp. 314-331

  
  

  Abstract

  The development of automatic techniques to port a substantial portion of FEFLO, a general‐purpose legacy CFD code operating on unstructured grids, to run on GPUs is described. FEFLO is a typical adaptive, edge‐based finite element code for the solution of compressible and incompressible flows, which is primarily written in Fortran 77 and has previously been ported to vector, shared memory parallel and distributed memory parallel machines. Owing to the large size of FEFLO and the likelihood of human error in porting, as well as the desire for continued development within a single codebase, a specialized Python script, based on FParser (Int. J. Comput. Sci. Eng. 2009; 4 :296–305), was written to perform automated translation from the OpenMP‐parallelized edge and point loops to GPU kernels implemented in CUDA, along with GPU memory management. The results of verification benchmarks and performance indicate that performances achieved by such a translator can rival those of codes rewritten by specialists. The approach should be of general interest, as how best to run on GPUs is being presently considered for many so‐called legacy codes.

  Abstract
  The development of automatic techniques to port a substantial portion of FEFLO, a general‐purpose legacy CFD code operating on unstructured grids, to run on GPUs is described. [...]

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• Numerical Studies of Green Water Impact on Fixed and Moving Bodies

  H. Lu, C. Yang, R. Lohner

  International Journal of Offshore and Polar Engineering (2012). Vol. 22 (2), pp. 123-132

  
  

  Abstract

  A numerical time domain simulation model has been developed to study green water phenomena and its impact loading on structures. In this model, a volume-of-fluid (VOF) technique is used to capture the violent free-surface motion. The incompressible Euler/Navier-Stokes equations, written in an arbitrary Lagrangian-Eulerian (ALE) frame, are solved using projection schemes and a finite element method on unstructured grids. Numerical simulations of green water problems carried out in this study include: green water overtopping a fixed 2D deck; green water impact on a fixed 3D body with or without a vertical wall on the deck; and green water impact on the deck and deckhouse of a moving floating production, storage and offloading (FPSO) model. Numerical results obtained using the present model are compared with experimental measurements for each case; and very good qualitative and relatively good quantitative agreements are obtained. The present numerical model can be used for simulating green water effects.

  Abstract
  A numerical time domain simulation model has been developed to study green water phenomena and its impact loading on structures. In this model, a volume-of-fluid (VOF) technique [...]

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• Experimental and numerical analysis of a sphere falling into a viscous fluid

  M. Cruchaga, R. Lohner, D. Celentano

  Int. J. Numer. Meth. Fluids (2012). Vol. 69 (9), pp. 1496-1521

  
  

  Abstract

  The experimental and numerical analysis of spheres falling into viscous flows is considered. The physical model is built using a set of silicone and glass spheres falling into oil and water. The rigid‐body trajectory of the sphere and the free surface evolution are obtained from videos. The numerical results are obtained using two different finite element codes. The first code uses a fractional step approach with adaptive meshes and time‐step sizes whereas the second code uses a monolithic fully coupled fixed‐mesh technique. The results exhibit a good comparison between both numerical techniques and with the experiments.

  Abstract
  The experimental and numerical analysis of spheres falling into viscous flows is considered. The physical model is built using a set of silicone and glass spheres falling [...]

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• Semi‐automatic porting of a large‐scale CFD code to multi‐graphics processing unit clusters

  A. Corrigan, R. Lohner

  Int. J. Numer. Meth. Fluids (2011). Vol. 69 (11), pp. 1786-1796

  
  

  Abstract

  A typical large‐scale CFD code based on adaptive, edge‐based finite‐element formulations for the solution of compressible and incompressible flow is taken as a test bed to port such codes to graphics hardware (graphics processing units, GPUs) using semi‐automatic techniques. In previous work, a GPU version of this code was presented, in which, for many run configurations, all mesh‐sized loops required throughout time stepping were ported. This approach simultaneously achieves the fine‐grained parallelism required to fully exploit the capabilities of many‐core GPUs, completely avoids the crippling bottleneck of GPU–CPU data transfer, and uses a transposed memory layout to meet the distinct memory access requirements posed by GPUs. The present work describes the next step of this porting effort, namely to integrate GPU‐based, fine‐grained parallelism with Message‐Passing‐Interface‐based, coarse‐grained parallelism, in order to achieve a code capable of running on multi‐GPU clusters. This is carried out in a semi‐automated fashion: the existing Fortran–Message Passing Interface code is preserved, with the translator inserting data transfer calls as required. Performance benchmarks indicate up to a factor of 2 performance advantage of the NVIDIA Tesla M2050 GPU (Santa Clara, CA, USA) over the six‐core Intel Xeon X5670 CPU (Santa Clara, CA, USA), for certain run configurations. In addition, good scalability is observed when running across multiple GPUs. The approach should be of general interest, as how best to run on GPUs is being presently considered for many so‐called legacy codes.

  Abstract
  A typical large‐scale CFD code based on adaptive, edge‐based finite‐element formulations for the solution of compressible and incompressible flow is taken as a test [...]

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• Iterative solution applied to the Helmholtz equation: Complex deflation on unstructured grids

  R. Aubry, S. Dey, R. Lohner

  Comput. Methods Appl. Mech. Engrg., (2012). Vol. 241-244, pp. 155-171

  
  

  Abstract

  Extensions of deflation techniques developed for the Poisson and Navier equations (Aubry et al., 2008; Mut et al., 2010; Löhner et al., 2011; Aubry et al., 2011) [1], [2], [3], [4] are presented for the Helmholtz equation. Numerous difficulties arise compared to the previous case. After discretization, the matrix is now indefinite without Sommerfeld boundary conditions, or complex with them. It is generally symmetric complex but not Hermitian, discarding optimal short recurrences from an iterative solver viewpoint (Saad, 2003) [5]. Furthermore, the kernel of the operator in an infinite space typically does not belong to the discrete space. The choice of the deflation space is discussed, as well as the relationship between dispersion error and solver convergence. Similarly to the symmetric definite positive (SPD) case, subdomain deflation accelerates convergence if the low frequency eigenmodes are well described. However, the analytic eigenvectors are well represented only if the dispersion error is low. CPU savings are therefore restricted to a low to mid frequency regime compared to the mesh size, which could be still relevant from an application viewpoint, given the ease of implementation.

  Abstract
  Extensions of deflation techniques developed for the Poisson and Navier equations (Aubry et al., 2008; Mut et al., 2010; Löhner et al., 2011; Aubry et al., 2011) [1], [2], [3], [4] are [...]

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• Robust Boundary Layer Mesh Generation

  A. Loseille, R. Lohner

  (2013). Proceedings of the 21st International Meshing Roundtable. Springer, Berlin, Heidelberg

  
  

  Abstract

  In this paper, we introduce a 3D local operator that automatically combines typical simpler operators as removal of vertices, collapse of edges or swap of faces and edges. This operator is inherited from incremental methods where the mesh HkHk is modified iteratively through sequences of insertion of a point P: Hk+1=Hk−CP+BP,Hk+1=Hk−CP+BP, where CPCP is the cavity of P and BPBP the ball of P. We derive two algorithms to compute CpCp. The first algorithm is tuned to be a fast point reprojection to the geometry even in the presence of a boundary layer mesh. The second one is tuned to generate boundary layer meshes for complex geometries. We show how quasi-structured elements can be enforced. In addition, enhancements as multi-normals can be incorporated in the process. Both operators can be used with surface and volume point while preserving a given geometry. They rely on the use on an existing initial volume mesh and always produce a valid 3D mesh on output.

  Abstract
  In this paper, we introduce a 3D local operator that automatically combines typical simpler operators as removal of vertices, collapse of edges or swap of faces and edges. [...]

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