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 Praetorius, Simon committed Dec 30, 2017 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ``````AMDiS {#mainpage} ===== The *Adaptive Multi-Dimensional Simulation Toolbox* (AMDiS) is implemented as a discretization module on top of the [Dune](https://dune-project.org) framework. Example ------- An AMDiS program consists of three main incredients: 1. A Problem class that holds all information necessary for assembling a linear system, see \ref ProblemStat. 2. Operators describing the (bi)linear-form of your PDE, see \ref operators. 3. Adaption-modules for the time- and space adaptive solution of the problem, see \ref Adaption. **Poisson equation:** The most simple elliptic PDE is the Poisson equation: \f{eqnarray*}{ -\Delta u &=& f(x),\quad\mbox{ in }\Omega \\ u &=& g(x),\quad\mbox{ on }\partial\Omega \f} where \f\$ f(x) \f\$ and \f\$ g(x) \f\$ are parameters describing the volume and boundary forces of the problem. The corresponding weak form of the equation reads: \f[ \langle \nabla v, \nabla u\rangle_\Omega = (f(x)\,v)_\Omega,\quad\forall v\in V_0 \f] with \f\$ u\in V_g \f\$. Thus, we need to define a grid (discretization of \f\$ \Omega \f\$) and a finite `````` Praetorius, Simon committed Jan 02, 2018 32 ``````element space (discretization of \f\$ V_0 \f\$ and \f\$ V_g \f\$). This cab be `````` Praetorius, Simon committed Dec 30, 2017 33 34 35 36 37 38 39 ``````provided as `Traits` parameter in the ProblemStat class: ~~~~~~~~~~~~~~~{.cpp} using Grid = Dune::AlbertaGrid; using Traits = LagrangeBasis; ~~~~~~~~~~~~~~~ `````` Praetorius, Simon committed Jan 02, 2018 40 ``````where `AlbertaGrid` defines a grid and `Traits` a Lagrange Finite-Element space with `````` Praetorius, Simon committed Dec 30, 2017 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 ``````local polynomial degree 1 on the elements. All AMDiS programs start with initialization of the library, using `AMDiS::init` ~~~~~~~~~~~~~~~{.cpp} using namespace AMDiS; int main(int argc, char** argv) { AMDiS::init(argc, argv); ProblemStat prob("name"); ~~~~~~~~~~~~~~~ The Problem class is initialized with a name, that is used as identifier in the parameter files. In order to initialize the Finite-Element space, the grid and all other parts of the problem class, call `initialize(Flag)` where `Flag` specifies what to initialize. For now, we initialize everything: `INIT_ALL`: ~~~~~~~~~~~~~~~{.cpp} prob.initialize(INIT_ALL); ~~~~~~~~~~~~~~~ Operators specify the (bi-)linear-form and the coefficient function in the term, see \ref operators for a list of possible types. The bilinear-form in the Poisson `````` Praetorius, Simon committed Jan 02, 2018 65 ``````equation consists of a second-order term with coefficient = 1 and the linear-form `````` Praetorius, Simon committed Dec 30, 2017 66 67 68 69 70 71 72 73 74 75 76 ``````includes the function \f\$ f(x)=-1 \f\$: ~~~~~~~~~~~~~~~{.cpp} auto opL = makeOperator(tag::gradtest_gradtrial{}, 1.0); prob.addMatrixOperator(opL, 0, 0); auto opF = makeOperator(tag::test{}, [](auto const& x) { return -1.0; }, 0); prob.addVectorOperator(opF, 0); ~~~~~~~~~~~~~~~ Boundary conditions, in the example above a Dirichlet condition, is specified by `````` Praetorius, Simon committed Jan 02, 2018 77 78 ``````defining a predicate for the boundary \f\$ \Gamma\subset\partial\Omega \f\$ and the values on the boundary \f\$ g(x) = 0 \f\$: `````` Praetorius, Simon committed Dec 30, 2017 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 `````` ~~~~~~~~~~~~~~~{.cpp} auto predicate = [](auto const& x){ return x[0] < 1.e-8 || x[1] < 1.e-8; }; prob.addDirichletBC(predicate, 0, 0, 0.0); ~~~~~~~~~~~~~~~ The final step is the assembling and solution of the linear system. (Maybe including grid adaption). This is realized using an \ref AdaptStationary class: ~~~~~~~~~~~~~~~{.cpp} AdaptInfo adaptInfo("adapt"); AdaptStationary adapt("adapt", prob); adapt.adapt(); // assemble and solve ~~~~~~~~~~~~~~~ Finally, finish the AMDiS program with `AMDiS::finish()`. `````` Praetorius, Simon committed Jan 02, 2018 96 ``````The complete program then reads: `````` Praetorius, Simon committed Dec 30, 2017 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 ``````~~~~~~~~~~~~~~~{.cpp} #include #include #include #include using namespace AMDiS; using Grid = Dune::AlbertaGrid; using Traits = LagrangeBasis; int main(int argc, char** argv) { AMDiS::init(argc, argv); ProblemStat prob("poisson"); prob.initialize(INIT_ALL); auto opL = makeOperator(tag::gradtest_gradtrial{}, 1.0); prob.addMatrixOperator(opL, 0, 0); auto opF = makeOperator(tag::test{}, [](auto const& x) { return -1.0; }, 0); prob.addVectorOperator(opF, 0); // set boundary condition auto predicate = [](auto const& x){ return x[0] < 1.e-8 || x[1] < 1.e-8; }; prob.addDirichletBC(predicate, 0, 0, 0.0); // assemble and solve AdaptInfo adaptInfo("adapt"); AdaptStationary adapt("adapt", prob); adapt.adapt(); AMDiS::finalize(); return 0; } ~~~~~~~~~~~~~~~``````