DOLFIN User Manual - FEniCS Project
DOLFIN User Manual - FEniCS Project DOLFIN User Manual - FEniCS Project
DOLFIN User ManualHoffman, Jansson, Logg, Wellsother tensor only involving integration of products of basis functions andtheir derivatives over the reference element.For efficiency in the computation of the element matrices and vectors, FFCprecomputes the tensors that are independent of the geometry of a certaincell.7.6 Assemble matrices and vectorsThe class FEM automates the assembly algorithm, constructing a linear systemof equations from a PDE, given in the form of a variational problem (7.2),with a bilinear form a(·, ·) and a linear form L(·).The classes BilinearForm and LinearForm are automatically generated byFFC, and to assemble the corresponding matrix and vector for the Poissonproblem (7.2) with source term f, we write:Poisson::BilinearForm a;Poisson::LinearForm L(f);Mesh mesh;Mat A;Vec b;FEM::assemble(a,L,A,b,mesh);In the assemble() function the element matrices and vectors are computedby calling the function eval() in the classes Bilinearform and Linearform.The eval() functions at a certain element in the assembly algorithm take asargument an AffineMap object, describing the mapping from the referenceelement to the actual element, by computing the Jacobian J of the mapping(also J −1 and det(J) are computed).48
DOLFIN User ManualHoffman, Jansson, Logg, Wells7.7 Specifying boundary conditions and dataThe boundary conditions are specified by defining a new subclass of the classBoundaryCondition, which we here will name MyBC:class MyBC : public BoundaryCondition{const BoundaryValue operator() (const Point& p){BoundaryValue value;if ( std::abs(p.x - 0.0) < DOLFIN_EPS ) value = 0.0;if ( std::abs(p.x - 1.0) < DOLFIN_EPS ) value = 0.0;if ( std::abs(p.y - 0.0) < DOLFIN_EPS ) value = 0.0;if ( std::abs(p.y - 1.0) < DOLFIN_EPS ) value = 0.0;if ( std::abs(p.z - 0.0) < DOLFIN_EPS ) value = 0.0;if ( std::abs(p.z - 1.0) < DOLFIN_EPS ) value = 0.0;return value;}};where we have assumed homogeneous Dirichlet boundary conditions for theunit cube. We only need to specify the boundary conditions explicitly onthe Dirichlet boundary. On the remaining part of the boundary, homogeneousNeumann boundary conditions are automatically imposed weakly bythe variational form.The boundary conditions are then imposed as an argument to the assemble()function:MyBC bc;FEM::assemble(a,L,A,b,mesh,bc);There is currently no easy way to impose non-homogeneous Neumann boundaryconditions or other combinations of boundary conditions. This will beadded to a future release of DOLFIN.49
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- Page 25 and 26: Chapter 3Linear algebra◮ Develope
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- Page 29 and 30: Chapter 4The mesh◮ Developer’s
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<strong>DOLFIN</strong> <strong>User</strong> <strong>Manual</strong>Hoffman, Jansson, Logg, Wells7.7 Specifying boundary conditions and dataThe boundary conditions are specified by defining a new subclass of the classBoundaryCondition, which we here will name MyBC:class MyBC : public BoundaryCondition{const BoundaryValue operator() (const Point& p){BoundaryValue value;if ( std::abs(p.x - 0.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;if ( std::abs(p.x - 1.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;if ( std::abs(p.y - 0.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;if ( std::abs(p.y - 1.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;if ( std::abs(p.z - 0.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;if ( std::abs(p.z - 1.0) < <strong>DOLFIN</strong>_EPS ) value = 0.0;return value;}};where we have assumed homogeneous Dirichlet boundary conditions for theunit cube. We only need to specify the boundary conditions explicitly onthe Dirichlet boundary. On the remaining part of the boundary, homogeneousNeumann boundary conditions are automatically imposed weakly bythe variational form.The boundary conditions are then imposed as an argument to the assemble()function:MyBC bc;FEM::assemble(a,L,A,b,mesh,bc);There is currently no easy way to impose non-homogeneous Neumann boundaryconditions or other combinations of boundary conditions. This will beadded to a future release of <strong>DOLFIN</strong>.49
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