A Hybrid Parallel Algorithm for Transforming Finite Element Functions from Adaptive to Cartesian Grids

2010 ◽  
pp. 48-61 ◽  
Author(s):  
Oliver Fortmeier ◽  
H. Martin Bücker
Keyword(s):  
Finite Element ◽  
Parallel Algorithm ◽  
Cartesian Grids

scholarly journals Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2972
Author(s):  
Saray Busto ◽  
Michael Dumbser ◽  
Laura Río-Martín
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Yield Stress ◽  
Finite Volume ◽  
Power Law ◽  
Source Terms ◽  
Cartesian Grids ◽  
Rans Equations ◽  
Edge Based ◽  
Finite Element Schemes

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

A Parallel Algorithm for Stiffness Matrix Decomposition Using Threadpool Method

2012 ◽  
Vol 594-597 ◽  
pp. 2880-2885
Author(s):  
Jun Tao Chen ◽  
Ming Xiao ◽  
Hui Bo Liu
Keyword(s):  
Finite Element ◽  
Parallel Algorithm ◽  
Stiffness Matrix ◽  
Decomposition Method ◽  
High Efficiency ◽  
Matrix Decomposition ◽  
One Dimensional ◽  
Numerical Tests ◽  
Element Simulation ◽  
Nested Loops

To shorten calculation time in finite element simulation by using multithreading computer, a parallel algorithm for stiffness matrix decomposition based on threadpool method is proposed. Firstly, a decomposition method of applicability to parallel computation is put forward by transferring the Cholesky's LLT method. Then, the threadplool is employed to generate multithreading for repeating use and the optimization is conducted considering load-balancing of each thread. Finally, numerical tests by using proposed algorithm in decomposition of one-dimensional array stored stiffness matrix are carried out on different calculation platforms with multi-processors. It is shown that the parallel algorithm can overcome the limitations of OpenMP when being applied in nested loops and is of high efficiency on stiffness matrix decomposition with low platform demands. The algorithm has explicit concept and minor programming difficulty and is applicable to solve problems caused by limitation of OpenMP in particular.

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