Zhoufang Xiao, Jianjun Chen*, Yao Zheng, JifaZhangBoundary Layer Mesh Generation Using Vector Fields...
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Boundary Layer Mesh Generation Using Vector Fields Computed by the Boundary Element Zhoufang Xiao, Jianjun Chen*, Yao Zheng, Jifa Zhang Center for Engineering & Scientific Computation, Zhejiang University, China *Corresponding author: [email protected] Step 2. Boundary layer mesh generation This step needs three user parameters, indicating the height of the first layer (denoted by h 0 hereafter), the expansion ratio of neighboring layers (denoted by β hereafter) and the allowed maximal number of layers (denoted by n l hereafter), respectively. According to these parameters, we could compute the marching distance at each front node. Meanwhile, the marching direction at each front node could be computed by the boundary integration equation (i.e., Equation 2). Once the marching directions and marching distances are determined at all front nodes, a layer of prismatic elements could then be created by connecting front nodes and their duals after front propagation. Repeating this front propagation procedure for at most n l times, we could then create semi-structured prismatic elements in the vicinity of viscous walls. Unstructured mesh generation If a symmetry plane is defined on the domain boundary, layered quadrilateral elements should have been created in the vicinity of common curves of the symmetry plane and viscous walls after Step 2. Therefore, the surface mesh of the symmetry plane, which are initially composed of triangular elements only, need be update to accommodate these quadrilateral elements. After that, we could collect the surface triangles depicting the remaining unmeshed volume region. These triangles include those located at boundaries with non-viscous wall types and those depicting the outmost boundary of boundary layer elements. With these surface triangles as the input, we finally employ a DT based mesher to fill unstructured tetrahedra in the domain enclosed by the input surface triangles. The employed DT mesher features itself with its capability to create a boundary constrained tetrahedral mesh robustly. This feature is a key for the success of this step, where a point-to-point conformity is required between unstructured tetrahedra and boundary layer elements. The overall meshing flowchart IMR Poster Session CESC Solution of the governing equation BL mesh generation Unstructured mesh generation Apply appropriate boundary conditions on different parts of ∂Ω according to the boundary conditions used for viscous flow computations (see Equation 1), and then compute Equation 1 by employing the BEM. Instead of creating a new boundary discretization for the BEM, we use the input surface triangulation for the BEM solution process to avoid additional consumptions of computing resources. As a result, a vector (i.e., u) and the flux of this vector (i.e., ∂u/∂n) could be obtained for each element of the input surface triangulation . Step 1. Solution of the Laplacian equation Basic idea Denote the problem domain by ∂Ω, and its boundary by ∂Ω. A vector field over Ω is governed by, (1) 0 0 ( ) .. ( )= ( ) ( )/ ( ) D N p p st p p p p p p Δ = ∈Ω ∈ ∂Ω ∂ ∂ = ∈ ∂Ω u 0 u u u n μ Here, Δ=(∂ 2 /∂x 2 , ∂ 2 /∂y 2 , ∂ 2 /∂z 2 ) is the Laplacian operator. ∂Ω D and ∂Ω N are the part of boundaries where Dirichlet and Neumann boundary conditions are applied, respectively, n is the unit outer normal vector at point p belonging to ∂Ω N , and u 0 (p) and µ 0 (p) are the distributions of initial values at points belonging to ∂Ω D and ∂Ω N , respectively. Basically, the solution of u x (p) at point p 0 could be computed by the boundary element method (BEM) with the following boundary integration equation, i.e., 0 0 0 () G(,) () [G( , ) () ] x x x p up pp up pp up ds ∂Ω ∂ ∂ = − ∂ ∂ n n Here, p∈∂Ω, ds p represents the area of the small surface patch in the vicinity of p, n is the unit normal vector at point p towards the exterior of domain Ω, and 0 0 1 ln | | for 2D problems 2 G( , ) 1 for 3D problems 4 | | 0 p p p p p p π π − − = − is the fundamental solution of the Laplacian equation. BL mesh generation Given a surface triangulation of the domain boundary, the following three steps output a prismatic hybrid mesh: A 2D example indicating the capability of our method to avoid global self-intersections by analyzing the field of |u|. (a) The field of |u|. Where the interface collides, the value of |u| approaches to zero. (b) The boundary layer mesh. (c) The overall hybrid mesh. (a) (b) (c) (a) (b) The hybrid mesh of F6 model. (a) A cut view of the hybrid mesh. (b) Local details of meshes near the engine (a) (b) Meshes of the F6 model: (a) A mesh generated by the proposed method. (b) A mesh generated by a commercial tool. (c) presents a quality comparison of mesh a and mesh b. (a) (b) The hybrid mesh used for simulating the flow structure induced by the rotation of fan blades. (a) Geometry definition. (b) A cut view of the hybrid volume mesh, which contains a structured part near boundaries and an unstructured part filling in the remaining region. (c) is a close-up view of the mesh shown in (b), where a few boundary layer elements are highlighted. (d) Viscous flow simulation results (RNG k-ε turbulence model; the fan blades rotate at a speed of 3000 rpm): Velocity vector field (back view). (c) (c) The rotation region (d)
Transcript of Zhoufang Xiao, Jianjun Chen*, Yao Zheng, JifaZhangBoundary Layer Mesh Generation Using Vector Fields...
Boundary Layer Mesh Generation Using Vector Fields Computed by the Boundary Element
Zhoufang Xiao, Jianjun Chen*, Yao Zheng, Jifa ZhangCenter for Engineering & Scientific Computation, Zhejiang University, China
*Corresponding author: [email protected]
Step 2. Boundary layer mesh generationThis step needs three user parameters, indicating theheight of the first layer (denoted by h0 hereafter), theexpansion ratio of neighboring layers (denoted by βhereafter) and the allowed maximal number of layers(denoted by nl hereafter), respectively. According tothese parameters, we could compute the marchingdistance at each front node. Meanwhile, the marchingdirection at each front node could be computed by theboundary integration equation (i.e., Equation 2). Oncethe marching directions and marching distances aredetermined at all front nodes, a layer of prismaticelements could then be created by connecting frontnodes and their duals after front propagation. Repeatingthis front propagation procedure for at most nl times, wecould then create semi-structured prismatic elements inthe vicinity of viscous walls.
Unstructured mesh generationIf a symmetry plane is defined on the domainboundary, layered quadrilateral elements should havebeen created in the vicinity of common curves of thesymmetry plane and viscous walls after Step 2.Therefore, the surface mesh of the symmetry plane,which are initially composed of triangular elementsonly, need be update to accommodate thesequadrilateral elements. After that, we could collect thesurface triangles depicting the remaining unmeshedvolume region. These triangles include those locatedat boundaries with non-viscous wall types and thosedepicting the outmost boundary of boundary layerelements. With these surface triangles as the input, wefinally employ a DT based mesher to fill unstructuredtetrahedra in the domain enclosed by the input surfacetriangles. The employed DT mesher features itselfwith its capability to create a boundary constrainedtetrahedral mesh robustly. This feature is a key for thesuccess of this step, where a point-to-point conformityis required between unstructured tetrahedra andboundary layer elements.
The overall meshing flowchart
IMR Poster SessionCESC
Solution of the governing equation BL mesh generation Unstructured mesh generation
Apply appropriate boundary conditions on differentparts of ∂Ω according to the boundary conditions usedfor viscous flow computations (see Equation 1), andthen compute Equation 1 by employing the BEM.Instead of creating a new boundary discretization for theBEM, we use the input surface triangulation for theBEM solution process to avoid additional consumptionsof computing resources. As a result, a vector (i.e., u) andthe flux of this vector (i.e., ∂u/∂n) could be obtained foreach element of the input surface triangulation .
Step 1. Solution of the Laplacian equation
Basic ideaDenote the problem domain by ∂Ω, and its boundaryby ∂Ω. A vector field over Ω is governed by,
(1)0
0
( ). . ( )= ( )
( ) / ( )D
N
p ps t p p p
p p p
Δ = ∈Ω ∈ ∂Ω ∂ ∂ = ∈ ∂Ω
u 0u u
u n μ
Here, Δ=(∂2/∂x2, ∂2/∂y2, ∂2/∂z2) is the Laplacianoperator. ∂ΩD and ∂ΩN are the part of boundarieswhere Dirichlet and Neumann boundary conditionsare applied, respectively, n is the unit outer normalvector at point p belonging to ∂ΩN, and u0(p) and µ0(p)are the distributions of initial values at pointsbelonging to ∂ΩD and ∂ΩN, respectively.Basically, the solution of ux(p) at point p0 could be
computed by the boundary element method (BEM)with the following boundary integration equation, i.e.,
00 0
( ) G( , )( ) [G( , ) ( ) ]xx x p
u p p pu p p p u p ds∂Ω
∂ ∂= −∂ ∂ n n
Here, p∈∂Ω, dsp represents the area of the smallsurface patch in the vicinity of p, n is the unit normalvector at point p towards the exterior of domain Ω,and
0
0
1 ln | | for 2D problems2G( , ) 1 for 3D problems4 | |
0
p pp p
p p
π
π
− −= −
is the fundamental solution of the Laplacian equation.
BL mesh generationGiven a surface triangulation of the domain boundary, the following three steps output a prismatic hybrid mesh: A 2D example indicating the capability of our method to avoid global self-intersections by analyzing the
field of |u|. (a) The field of |u|. Where the interface collides, the value of |u| approaches to zero. (b) Theboundary layer mesh. (c) The overall hybrid mesh.
(a) (b) (c)
(a) (b)The hybrid mesh of F6 model. (a) A cut view of the hybrid mesh. (b) Local details of meshes near the engine
(a) (b)Meshes of the F6 model: (a) A mesh generated by the proposed method. (b) A mesh generated by acommercial tool. (c) presents a quality comparison of mesh a and mesh b.
(a) (b)The hybrid mesh used for simulating the flow structure induced by the rotation of fan blades. (a) Geometrydefinition. (b) A cut view of the hybrid volume mesh, which contains a structured part near boundaries and anunstructured part filling in the remaining region. (c) is a close-up view of the mesh shown in (b), where a fewboundary layer elements are highlighted. (d) Viscous flow simulation results (RNG k-ε turbulence model; the fanblades rotate at a speed of 3000 rpm): Velocity vector field (back view).
(c)
(c)
The rotation region
(d)
