CAESAR -- An European Union project (ESPRIT). ⢠CAESAR is an acronym derived from Clusters of computationally intensive Applications for Engineering ...
ICOME 2015 Viscous Grid Generation by the Boundary Element Method Yao Zheng, Zhoufang Xiao, Jianjun Chen, Jianjing Zheng Center for Engineering and Scientific Computation, and School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang, 310027, PR China
October 11 – 13, 2015, Hangzhou, China
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Outline 1
Background Information
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Motivation
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PDE based Method
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Conclusions and Future Work
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Background Information
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CAESAR Project, 1995-1996 CAESAR -- An European Union project (ESPRIT) CAESAR is an acronym derived from Clusters of computationally intensive Applications for Engineering design and Simulation on scalable pARallel platforms. The CAESAR project enables and demonstrates the exploitation of HPCN (High Performance Computing and Networking) to facilitate the optimization of design and production cycles in the manufacturing industry as product complexity increases. A subtask of CAESAR is Parallel Simulation User Environment (PSUE), which was carried out at University of Wales Swansea, UK.
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Parallel Simulation User Environment A general schematic of the Parallel Simulation User Environment (PSUE)
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Parallel Simulation User Environment A selection of windows from the PSUE driving a typical user session
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Enabling Environment for Multidisciplinary Application Simulations (EEMAS), 2002 A snapshot of an EEMAS session
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EEMAS Architecture and Features The EEMAS architecture from users’ perspective
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HEDP (High End Digital Prototyping) System, 2005 A snapshot of a HEDP session running on an SGI Octane2 machine
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Mesh Generation Types of Meshes generated
Structured Meshes Unstructured Meshes Anisotropic Meshes Hybrid Meshes
Meshes Generated in parallel
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Anisotropic Meshes
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Structured Mesh Dominated Hybrid Meshes (DRAGON Grid). NASA Glenn, 1998 Film-cooled turbine vane
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The Entire Parallel Meshing Process #volume elements: about 80M tets ; #cores: 64 Overall timing cost for the entire parallel process: 8mn
F16 test model meshes generated in parallel
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A Parallel CFD Flowchart A research program was started to enhance our in-house CFD software capability so as to provide the necessary basis for the next generation of simulations. It was deemed necessary to parallelise all of the steps in the computational cycle, from unstructured meshing to simulation, to visualization and data mining, and even with mesh adaptation.
Parallel Mesher
Parallel Solver
Parallel Visualizer 14
Parallel Simulation of Moving Boundary Problems
The key technologies: parallel adaptive meshing; parallel local remeshing.
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23rd International Meshing Roundtable, IMR 2014 The winner for the meshing contest in the 23rd International Meshing Roundtable (IMR23) hold in London, October 2014. The IMR is one of the most renowned conferences in the community of mesh generation. It was initialised by Sandia National Laboratory in 1992 and has been hold once every year since then. Because IMR23 was hold in London, a London Tower Bridge model is selected as the benchmark geometry for meshing contest.
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London Tower Bridge (Picture)
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London Tower Bridge (The Geometry)
The input STEP file contains 3,018 NURBS surfaces and 10,186 NURBS curves. 18
The Flow Induced by a Crosswind (v=34m/s)
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All in One
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Towards Extreme Scale Computing Sequential meshing
Parallel analysis
Parallel meshing
Parallel analysis
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Towards Extreme Scale Computing
国家超级计算广州中心 National Supercomputer Center in Guangzhou
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Ongoing Research
Groups of Birds
Aerodynamic optimization of a whole civil airplane
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Motivation
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Complexities of Computational Fluid Dynamics Sophisticated physics: The main features that are encountered in flow fields include boundary layers, wakes, shock waves and vortices.
In the flight test of F15, quiet spike is installed at the aircraft head to reduce the sonic boom.
The flow field around this region is extreme anisotropic (The iso-Mach line is shown in the figure).
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Complexities of Computational Fluid Dynamics Sophisticated physics: The main features that are encountered in flow fields include boundary layers, wakes, shock waves and vortices.
The simulation results of three dimensional liftoff flames
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Complexities of Computational Fluid Dynamics Complex geometries: The computational geometries often involve many features with different scales, sometimes moving boundaries are included.
Simulation of moving boundary problems (F16 Fighter)
The computational mesh of F16 and the detail features of its gears 27
Importance of Meshing to the Successful CFD Mesh quality determines the accuracy & efficiency of CFD computation The mesh generation step consumes most of the wall time during the whole process of analysis. Professor Baker from Princeton Univ. made a comparison of different types of meshes used in RANS computation:
good meshes lie on the diagonal line.
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Boundary Layer Meshes Are Needed The gradients of physical quantities tend to be great in the region near walls, high aspect ratio and thin boundary layer meshes can ensure the accuracy and efficiency of the result solution.
The use of hybrid meshes, composed of boundary layer mesh near walls and unstructured mesh in far field, can balance the viscous accuracy and easy of use.
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Methods of Boundary Layer Mesh Generation ALM (Advancing Layer Method): propagating each node at the front along its normal vector, new elements are formed with neighboring nodes.
ALM
Checking the validity of a normal vector
The normal vector at each node is determined by averaging the unit normal vectors of the faces sharing the node. 30
Methods of Boundary Layer Mesh Generation ALM: Depends on local geometric heuristics, and will fail to handle three-dimensional complex geometries.
√
？
Intensive artificial interactions are needed to generate boundary layer meshes for complex geometries. Even so we can not get satisfactory results.
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Methods of Boundary Layer Mesh Generation Problems encountered
Local intersections
Global intersections
A complex corner node
How to detect and resolve self-intersections and how to calculate right normal vectors at complex corner nodes?
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Methods of Boundary layer mesh generation AFLR (Advancing Front / Local Reconnect): generating anisotropic triangular / tetrahedral elements near walls, then locally reconnect such elements to form quads / prisms.
This approach is used in NASA VGRID / Pointwise T-REX Difficulties： For degenerated cases: some triangular / tetrahedral elements are unable to merge
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PDE Based Method
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Main Idea of PDE Based Method Main idea：Try to solve a field over the domain needed to be discretized, and the field is governed by a PDE and subject to some boundary conditions. FEM / BEM are used to solve such a field, and grid points of each layer are placed on iso-lines / iso-surfaces.
Iso-lines of DLR F6
Iso-surfaces of DLR F6 35
Advantages of PDE Based Method The advantages of the PDE based method are:
Intersections can be avoided; Right normal vectors can be calculated. Local intersections
Global intersections
A complex corner node
The front The boundary 36
Eikonal Equation Based Method Eikonal equation-based: A distance field Φ governed by the following equation,
∂φ 2 ∂φ 2 ∂φ 2 ( ) +( ) +( ) = 1 ∂x ∂y ∂z = s.t. φ 0, if p ∈ ∂Ω
Here, p is an arbitrary point in the computational space Ω (Ω covers ∂Ω), Φ is the minimum Euclidean distance from p to ∂Ω, and Φ can also be viewed as offset distance in front propagation problem.
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Eikonal Equation Based Method Eikonal equation-based:
The boundary The mesh
The iso-distance lines
The calculation of distance field adopts FEM or the Fast Sweeping Scheme
Need to discretize the domain, Meshes near walls should be finer to obtain accurate results. 38
Our Method: An Intuitive Idea Try to solve a steady heat conduction problem governed by Laplace equation, the each layer grid points are placed on isothermal surfaces. BEM are used to solve such a thermal field
No need to mesh the domain; Accurate normal can be obtained at each node.
Iso-lines of DLR F6
Iso-surfaces of DLR F6
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Our Method: Governing Equation A thermal field u governed by the following equation:
∂ 2u ∂ 2u ∂ 2u + 2+ 2 = 0 2 ∂x 0 ∂x1 ∂x 2 Subject to either Dirichlet or Neumann conditions at ∂Ω , that is,
∂u ( x) = u ( x) u0 (x), x ∈ ∂Ω or = q0 ( x), x ∈ ∂Ω ∂n Here Ω denotes the problem domain, and its boundary is ∂Ω.
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Our Method: Boundary Integral Equation The field value u(y) at an arbitrary point y is calculated by the boundary integral equation, = u (y )
Here q(x) =
∂u (x) ∂n x
∫
, and
Γ
[G(x, y ) q(x) − u(x)
G(x, y )
∂ G(x, y ) ]dsx ∂n x
is a fundamental solution, which is given by
1 1 ln 2π | x − y | , G(x, y ) = 1 1 , 4π | x − y |
for 2 D problem for 3D problem
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Our Method: Boundary Integral Equation In BEM, only boundaries are needed to be discretized. And the field values of thermal problem are calculated by accurate integral equations. The discretized form of the previous equation is, ∂ G(x, y ) K u (y ) ∑ ∫ [G(x, y )∑ N k (ξ )qk − N k (ξ ) u k ]dse ∑ Γe ∂n x k 1 =e 1 = = k 1 NE
K
And the gradient, ∂u (y ) NE ∂ G(x, y ) K ∂ G(x, y ) K [ N k (ξ )qk − N k (ξ ) u k ]dse ∑ ∑ ∑ ∫ Γe ∂ ∂ ∂ ∂ y y y n = = e 1= k 1 x i i i k 1
Then the normal vector at a point p is, np = (
∂u (y ) ∂u (y ) ∂u (y ) , , ) ∂y0 ∂y1 ∂y2
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Our Method: Preliminary Results Results
Works ? Probably YES
A boundary layer mesh of quads
A cut view of the boundary layer mesh of a box
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Our Method: Challenges There are problems when proximity features exist
Works ? Actually NO
The thermal values in the region around proximity features barely change
NOT suitable for boundary layer mesh generation 44
Measures Could Be Taken (1) .
Measures: 1. Set artificial boundaries 2. Set heat sources The governing equation changes to a Poisson equation,
∂ 2u ∂ 2u ∂ 2u + 2+ 2 = v( x0 , x1 , x2 ) 2 ∂x 0 ∂x1 ∂x 2
Artificial boundaries It seems difficult to set artificial boundaries and heat sources for complex geometries 45
Measures Could Be Taken (2) .
An alternative way (Method 3): change the scalar function u(x) to a vector function u(x) with components (u0 , u1 , u2 ) , and the governing equation becomes,
∂ 2u ∂ 2u ∂ 2u + 2 + 2 = 0 2 ∂x 0 ∂x1 ∂x 2 The three components of the vector equation, ∂ 2 u0 ∂ 2 u0 ∂ 2 u0 0 2 + 2 + 2 = ∂x1 ∂x 2 ∂x 0 2 2 2 ∂ u1 ∂ u1 ∂ u1 0 2 + 2 + 2 = x x x ∂ ∂ ∂ 1 2 0 2 2 2 ∂ u ∂ u ∂ u2 2 2+ 0 + = 2 2 2 ∂x 0 ∂x1 ∂x 2 46
Our Method (Method 3) .
The Dirichlet boundary condition at a point p of ∂Ω is defined as the normal vector (its magnitude is set to |u(x)|) up p
Following the computation of u(x) , the quantity Φ is derived, where
Φ =| u(x) |
Then the scalar field Φ is obtained, and the normal vector at an arbitrary point in the field is n(u0 , u1 , u2 ).
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The Flowchart of the Present Method .
The flowchart of the algorithm is as follows: Discretise boundaries. Initial front points refer to those mesh points on wall boundaries. Set BCS. Set the far field boundaries and wall boundaries with Dirichlet conditions, and set the symmetry boundaries with Neumann conditions; Propagate. Calculate the normal vector at each front point, and propagate these points along their normal vectors to new positions. Create elements. Connect the newly generated points with the previous front points to form a new layer meshes. Repeat. Go to Step 3 until either of the two stopping criteria is satisfied for each front point: The layer number of the point is equal to the prescribed total boundary layers; The normal vector at the front point is inverted compared to the normal vector at its previous front point. 48
The Results of the Present Method No artificial boundaries are needed
The scalar field Φ
The iso-lines
The boundary layer mesh and its close-up view
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The Results of the Present Method
The scalar field Φ of an airfoil with three components
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The Results of the Present Method
(b)
(c)
(d) (a)
The mesh of an airfoil with three components
(a)
(b)
(c)
(d)
The close-up view of the mesh of an airfoil with three components
Direction enrichment and direction collapse are introduced around the convex and concave corners, respectively
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The Results of the Present Method
The mesh of DLR F6 model 52
The Results of the Present Method
The close-up view of the mesh of DLR F6 model
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Conclusions and Future Work
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Concluding Remarks HEDP is introduced first, it is a platform for simulation including several components, i.e., Mesh generation, Solver and
Visualization A novel PDE based approach is proposed for boundary layer mesh generation A scalar field is calculated over the domain, the field is governed by a Laplace equation BEM is used for the calculation the field, which ensures the
accuracy of the solution and avoids the generation of a volume grid During the generation of boundary layer mesh, intersections can be
avoided and right normal vectors can be obtained 55
Future Work Parallelize the boundary layer mesh generation process BEM solver Consumes most of the time (over 90%) Mesh generation No need to be parallelized Develop local update schemes for boundary layer meshes oriented to moving boundary problems Update unstructured meshes only Update unstructured meshes and boundary layer meshes simultaneously
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Thank you！
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