Introduction And Basic Implementation For Finite Element .
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionMathematical Foundation of Finite ElementMethodsChapter 2: 2D/3D Finite Element SpacesXiaoming HeDepartment of Mathematics & StatisticsMissouri University of Science & Technology1 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionOutline12D uniform Mesh2Triangular elements3Rectangular elements43D elements5More discussion2 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionOutline12D uniform Mesh2Triangular elements3Rectangular elements43D elements5More discussion3 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: uniform partitionConsider Ω [left, right] [bottom, top].4 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: uniform partitionConsider Ω [left, right] [bottom, top].First, we form a uniform rectangular partition of Ω into N1elements in x axis and N2 elements in y axis with meshsizeright left top bottomh [h1 , h2 ] [,].N1N24 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesFor example, when N1 N2 8, we have5 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesThen we divide each rectangular element into two triangularelements by connecting the left-top corner and the right-lowercorner of the rectangular element.For example, when N1 N2 8, we have6 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesThis would give an uniform triangular partition.7 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesThis would give an uniform triangular partition.There are N 2N1 N2 elements and Nm (N1 1)(N2 1)mesh nodes.7 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and mesh nodes Zk (k 1, · · · , Nm ).8 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: global indicesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and mesh nodes Zk (k 1, · · · , Nm ).For example, when N1 N2 2, we have369483725826115478 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: local node indexLet Nl denote the number of local mesh nodes in a meshelement. Define your index for the local mesh nodes in a meshelement.3112329 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.10 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.Define matrix T to be an information matrix consisting of theglobal node indices of the mesh nodes of all the meshelements.10 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.Define matrix T to be an information matrix consisting of theglobal node indices of the mesh nodes of all the meshelements.We can use the j th column of the matrix P to store thecoordinates of the j th mesh node and the nth column of thematrix T to store the global node indices of the mesh nodesof the nth mesh element. For example, when N1 N2 2, wehave10 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.Define matrix T to be an information matrix consisting of theglobal node indices of the mesh nodes of all the meshelements.We can use the j th column of the matrix P to store thecoordinates of the j th mesh node and the nth column of thematrix T to store the global node indices of the mesh nodesof the nth mesh element. For example, when N1 N2 2, wehave 0 0 0 0.5 0.5 0.5 1 1 1P ,0 0.5 1 0 0.5 1 0 0.5 1 1 2 2 3 4 5 5 6T 4 4 5 5 7 7 8 8 .2 5 3 6 5 8 6 910 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge indexDefine your index for the boundary edges.11 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge indexDefine your index for the boundary edges.For example, when N1 N2 2, we have6574831211 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .boundaryedges(4, k) is the global node index of the secondend node of the k th boundary boundary edge ek .12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .boundaryedges(4, k) is the global node index of the secondend node of the k th boundary boundary edge ek .Set nbe size(boundaryedges, 2) to be the number ofboundary edges;12 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixFor the mesh with N1 N2 2 and all Dirichlet boundarycondition, we have:13 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular mesh: boundary edge information matrixFor the mesh with N1 N2 2 and all Dirichlet boundarycondition, we have: 1 1 1 1 1 1 1 1 15688431boundaryedges 1478963247896321 . 13 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular meshWhat are the information matricesP, T , boundaryedgesfor the following mesh?14 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionTriangular meshWhat are the information matricesP, T , boundaryedgesfor a general uniform triangular mesh with the mesh sizeh [h1 , h2 ] [right left top bottom,]N1N2in the domainΩ [left, right] [bottom, top]?15 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: uniform partitionConsider Ω [left, right] [bottom, top].16 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: uniform partitionConsider Ω [left, right] [bottom, top].Consider a uniform rectangular partition of Ω into N1 elementsin x axis and N2 elements in y axis with mesh sizeh [h1 , h2 ] [right left top bottom,].N1N216 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: uniform partitionConsider Ω [left, right] [bottom, top].Consider a uniform rectangular partition of Ω into N1 elementsin x axis and N2 elements in y axis with mesh sizeh [h1 , h2 ] [right left top bottom,].N1N2There are N N1 N2 elements and Nm (N1 1)(N2 1)mesh nodes.16 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: uniform partitionFor example, when N1 N2 8, we have17 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: global indicesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and mesh nodes Zk (k 1, · · · , Nm ).18 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: global indicesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and mesh nodes Zk (k 1, · · · , Nm ).For example, when N1 N2 2, we have362245119834718 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: local node indexLet Nl denote the number of local mesh nodes in a meshelement. Define your index for the local mesh nodes in a meshelement.431219 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.20 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.Define matrix T to be an information matrix consisting of theglobal node indices of the mesh nodes of all the meshelements.20 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: information matricesDefine matrix P to be an information matrix consisting of thecoordinates of all mesh nodes.Define matrix T to be an information matrix consisting of theglobal node indices of the mesh nodes of all the meshelements.For example, when 0P 0 1 4T 52N1 N2 2, we have0 0 0.5 0.5 0.5 1 1 10.5 1 0 0.5 1 0 0.5 1 2 4 55 7 8 .6 8 9 3 5 6 ,20 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge indexDefine your index for the boundary edges.21 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge indexDefine your index for the boundary edges.For example, when N1 N2 2, we have6574831221 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .boundaryedges(4, k) is the global node index of the secondend node of the k th boundary boundary edge ek .22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixMatrix boundaryedges:boundaryedges(1, k) is the type of the k th boundary edge ek :Dirichlet (-1), Neumann (-2), Robin (-3).boundaryedges(2, k) is the index of the element whichcontains the k th boundary edge ek .Each boundary edge has two end nodes. We index them asthe first and the second counterclock wise along the boundary.boundaryedges(3, k) is the global node index of the first endnode of the k th boundary boundary edge ek .boundaryedges(4, k) is the global node index of the secondend node of the k th boundary boundary edge ek .Set nbe size(boundaryedges, 2) to be the number ofboundary edges;22 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixFor example, when N1 N2 2 and all the boundary areDirichlet type, we have:23 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular mesh: boundary edge information matrixFor example, when N1 N2 2 and all the boundary areDirichlet type, we have: 1 1 1 1 1 1 1 1 13344221boundaryedges 1478963247896321 . 23 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular meshWhat are the information matricesP, T , boundaryedgesfor the following mesh?24 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionRectangular meshWhat are the information matricesP, T , boundaryedgesfor a general uniform rectangular mesh with the mesh sizeh [h1 , h2 ] [right left top bottom,]N1N2in the domainΩ [left, right] [bottom, top]?25 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussionOutline12D uniform Mesh2Triangular elements3Rectangular elements43D elements5More discussion26 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThe “reference local global” framework will be used toconstruct the finite element spaces.27 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThe “reference local global” framework will be used toconstruct the finite element spaces.We only consider the nodal basis functions (Lagrange type) inthis course.27 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThe “reference local global” framework will be used toconstruct the finite element spaces.We only consider the nodal basis functions (Lagrange type) inthis course.We first consider the reference 2D linear basis functions onthe reference triangular element Ê 4Â1 Â2 Â3 whereÂ1 (0, 0), Â2 (1, 0), and Â3 (0, 1).27 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThe “reference local global” framework will be used toconstruct the finite element spaces.We only consider the nodal basis functions (Lagrange type) inthis course.We first consider the reference 2D linear basis functions onthe reference triangular element Ê 4Â1 Â2 Â3 whereÂ1 (0, 0), Â2 (1, 0), and Â3 (0, 1).Define three reference 2D linear basis functionsψ̂j (x̂, ŷ ) aj x̂ bj ŷ cj , j 1, 2, 3,27 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThe “reference local global” framework will be used toconstruct the finite element spaces.We only consider the nodal basis functions (Lagrange type) inthis course.We first consider the reference 2D linear basis functions onthe reference triangular element Ê 4Â1 Â2 Â3 whereÂ1 (0, 0), Â2 (1, 0), and Â3 (0, 1).Define three reference 2D linear basis functionsψ̂j (x̂, ŷ ) aj x̂ bj ŷ cj , j 1, 2, 3,such that ψ̂j (Âi ) δij 0,1,if j 6 i,if j i,for i, j 1, 2, 3.27 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThen it’s easy to obtainψ̂1 (Â1 ) 1 c1 1,28 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThen it’s easy to obtainψ̂1 (Â1 ) 1 c1 1,ψ̂1 (Â2 ) 0 a1 c1 0,28 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThen it’s easy to obtainψ̂1 (Â1 ) 1 c1 1,ψ̂1 (Â2 ) 0 a1 c1 0,ψ̂1 (Â3 ) 0 b1 c1 0,28 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThen it’s easy to obtainψ̂1 (Â1 ) 1 c1 1,ψ̂1 (Â2 ) 0 a1 c1 0,ψ̂1 (Â3 ) 0 b1 c1 0,ψ̂2 (Â1 ) 0 c2 0,ψ̂2 (Â2 ) 1 a2 c2 1,ψ̂2 (Â3 ) 0 b2 c2 0,28 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsThen it’s easy to obtainψ̂1 (Â1 ) 1 c1 1,ψ̂1 (Â2 ) 0 a1 c1 0,ψ̂1 (Â3 ) 0 b1 c1 0,ψ̂2 (Â1 ) 0 c2 0,ψ̂2 (Â2 ) 1 a2 c2 1,ψ̂2 (Â3 ) 0 b2 c2 0,ψ̂3 (Â1 ) 0 c3 0,ψ̂3 (Â2 ) 0 a3 c3 0,ψ̂3 (Â3 ) 1 b3 c3 1.28 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsHencea1 1, b1 1, c1 1,a2 1, b2 0, c2 0,a3 0, b3 1, c3 0.29 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsHencea1 1, b1 1, c1 1,a2 1, b2 0, c2 0,a3 0, b3 1, c3 0.Then the three reference 2D linear basis functions areψ̂1 (x̂, ŷ ) x̂ ŷ 1,ψ̂2 (x̂, ŷ ) x̂,ψ̂3 (x̂, ŷ ) ŷ .29 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: reference basis functionsPlots of the three linear basis functions on the referencetriangle:30 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: local node indexLet Nlb denote the number of local finite element nodes (localfinite element basis functions) in a mesh element. HereNlb 3. Define your index for the local finite element nodesin a mesh element.31123231 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesThe mesh information matrices P and T are for the meshnodes.32 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesThe mesh information matrices P and T are for the meshnodes.We also need similar finite element information matrices Pband Tb for the finite elements nodes, which are the nodescorresponding to the finite element basis functions.32 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesThe mesh information matrices P and T are for the meshnodes.We also need similar finite element information matrices Pband Tb for the finite elements nodes, which are the nodescorresponding to the finite element basis functions.Note: For the nodal finite element basis functions, thecorrespondence between the finite elements nodes and thefinite element basis functions is one-to-one in astraightforward way. But it could be more complicated forother types of finite element basis functions in the future.32 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesThe mesh information matrices P and T are for the meshnodes.We also need similar finite element information matrices Pband Tb for the finite elements nodes, which are the nodescorresponding to the finite element basis functions.Note: For the nodal finite element basis functions, thecorrespondence between the finite elements nodes and thefinite element basis functions is one-to-one in astraightforward way. But it could be more complicated forother types of finite element basis functions in the future.Let Nb denote the total number of the finite element basisfunctions ( the number of unknowns the total number ofthe finite element nodes). Here Nb Nm (N1 1)(N2 1).32 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and finite element nodesXj (j 1, · · · , Nb ) (or the finite element basis functions).33 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesDefine your global indices for all the mesh elementsEn (n 1, · · · , N) and finite element nodesXj (j 1, · · · , Nb ) (or the finite element basis functions).For example, when N1 N2 2, we have3694837258261154733 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesDefine matrix Pb to be an information matrix consisting ofthe coordinates of all finite element nodes.34 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesDefine matrix Pb to be an information matrix consisting ofthe coordinates of all finite element nodes.Define matrix Tb to be an information matrix consisting ofthe global node indices of the finite element nodes of all themesh elements.34 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: information matricesFor the 2D linear finite elements, Pb and Tb are the same asthe P and T of the triangular mesh since the nodes of the 2Dlinear finite element basis functions are the same as those ofthe mesh. For example, when N1 N2 2, we have 0 0 0 0.5 0.5 0.5 1 1 1Pb P ,0 0.5 1 0 0.5 1 0 0.5 1 1 2 2 3 4 5 5 6Tb T 4 4 5 5 7 7 8 8 .2 5 3 6 5 8 6 935 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node indexDefine your index for the boundary finite element nodes.36 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node indexDefine your index for the boundary finite element nodes.For example, when N1 N2 2, we have,7658412336 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).The intersection nodes of Dirichlet boundary condition andother boundary conditions usually need to be treated asDirichlet boundary nodes.37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).The intersection nodes of Dirichlet boundary condition andother boundary conditions usually need to be treated asDirichlet boundary nodes.boundarynodes(2, k) is the global node index of the k thboundary boundary finite element node.37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).The intersection nodes of Dirichlet boundary condition andother boundary conditions usually need to be treated asDirichlet boundary nodes.boundarynodes(2, k) is the global node index of the k thboundary boundary finite element node.Set nbn size(boundarynodes, 2) to be the number ofboundary finite element nodes;37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).The intersection nodes of Dirichlet boundary condition andother boundary conditions usually need to be treated asDirichlet boundary nodes.boundarynodes(2, k) is the global node index of the k thboundary boundary finite element node.Set nbn size(boundarynodes, 2) to be the number ofboundary finite element nodes;For the above example with all Dirichlet boundary condition,we have:37 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: boundary node information matrixMatrix boundarynodes:boundarynodes(1, k) is the type of the k th boundary finiteelement node: Dirichlet (-1), Neumann (-2), Robin (-3).The intersection nodes of Dirichlet boundary condition andother boundary conditions usually need to be treated asDirichlet boundary nodes.boundarynodes(2, k) is the global node index of the k thboundary boundary finite element node.Set nbn size(boundarynodes, 2) to be the number ofboundary finite element nodes;For the above example with all Dirichlet boundary condition,we have: 1 1 1 1 1 1 1 1boundarynodes .1478963237 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: affine mappingNow we can use the affine mapping between an arbitrarytriangle E 4A1 A2 A3 and the reference triangleÊ 4Â1 Â2 Â3 to construct the local basis functions from thereference ones.38 / 100
2D uniform MeshTriangular elementsRectangular elements3D elementsMore discussion2D linear finite element: a
logic relationship between the 1D element index (the nth element) and the 1D global node indices of the vertices of the elements, through the 2D element index (the natural \row" index r e and \column" index c e of an element in the 2D mesh) and the 2D node index (the natural \row" index r n and \column" index c n of a node in the 2D mesh) 11/103
Bruksanvisning för bilstereo . Bruksanvisning for bilstereo . Instrukcja obsługi samochodowego odtwarzacza stereo . Operating Instructions for Car Stereo . 610-104 . SV . Bruksanvisning i original
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This presentation and SAP's strategy and possible future developments are subject to change and may be changed by SAP at any time for any reason without notice. This document is 7 provided without a warranty of any kind, either express or implied, including but not limited to, the implied warranties of merchantability, fitness for a .
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Den kanadensiska språkvetaren Jim Cummins har visat i sin forskning från år 1979 att det kan ta 1 till 3 år för att lära sig ett vardagsspråk och mellan 5 till 7 år för att behärska ett akademiskt språk.4 Han införde två begrepp för att beskriva elevernas språkliga kompetens: BI
