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 Praetorius, Simon committed Apr 14, 2020 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 32 33 34 35 36 37 # GridFunctions {: #group-gridfunctions} ## Summary GridFunctions are objects that can be evaluated in global coordinates and can be restricted to grid elements and evaluated there in local coordinates. c++ ProblemStat prob("name"); prob.initialize(INIT_ALL); // create a grid function auto gridFct = makeGridFunction(EXPRESSION, prob.gridView()); // eval GridFunction at global coordinates auto global = Dune::FieldVector{1.0, 2.0}; auto value = gridFct(global); // create a local representation of the grid function auto localFct = localFunction(gridFct); for (auto const& element : elements(prob.gridView())) { // bind the local function to a grid element localFct.bind(element); // eval LocalFunction at local coordinates auto local = element.geometry().center(); value = localFct(local); localFct.unbind(); }  GridFunctions are build from expressions that are a composition of some elementary terms. ### Examples of expressions Before we give examples where and how to use GridFunctions, we demonstrate what an Expression could be, to create a GridFunction from. In the following examples,  Müller, Felix committed Sep 29, 2020 38 we assume that a [ProblemStat](../Problem#class-problemstat) named prob is already  Praetorius, Simon committed Apr 14, 2020 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 created and initialized. #### 1. Discrete Functions Components of a solution vector are discrete functions, i.e. grid function build by a linear combination of local basis functions with a vector of coefficients. c++ auto expr1 = prob.solution(0);  #### 2. Constants and functions Constant values (or reference to values) and functions of global coordinates are expressions c++ auto expr2 = 1.0; double var = 1.0; auto expr3 = std::ref(var); auto expr4 = [](auto const& x) { return x[0] + x[1]; };  !!! warning Combining std::ref and constant expressions could result in unexpected behavior. Note that a reference wrapper implicitly converts to the underlying data type and thus std::ref + double results in double and the reference is gone. Use a lambda function and a function wrapper instead. #### 3. Composition of expressions Combining expressions using arithmetic operations or some mathematical functions build a new expressions: c++ auto expr5 = prob.solution(0) + 1.0; auto expr6 = max(evalAtQP(expr4), expr3);  This works, if at least one of the expressions involved is already a grid functions, so one can not simply add two lambda functions c++ // ERROR: auto no_expr = expr4 + 1.0;  ### Examples of usage In the following examples, an Expression is anything a GridFunction can be created from, sometimes also called PreGridFunction. It includes constants, functors callable with global coordinates, and any combination of GridFunctions. #### 1. Usage of GridFunctions to build Operators: c++ ProblemStat prob("name"); prob.initialize(INIT_ALL); auto opB = makeOperator(BiLinearForm, Expression); prob.addMatrixOperator(opB, Row, Col); auto opL = makeOperator(LinearForm, Expression); prob.addVectorOperator(opL, Row);   Müller, Felix committed Sep 29, 2020 98 See also [makeOperator()](../Operators#function-makeoperator).  Praetorius, Simon committed Apr 14, 2020 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154  #### 2. Usage of GridFunctions in BoundaryConditions: c++ prob.addDirichletBC(Predicate, Row, Col, Expression);  #### 3. Interpolate a GridFunction to a DOFVector: c++ prob.solution(_0).interpol(Expression); prob.solution() << Expression;  #### 4. Integrate a GridFunction on a GridView: c++ auto value = integrate(Expression, prob.gridView());  ## List of Grid Functions and Expressions Function | Descriptions -------------------------------------------|--------------------------------------------- [evalAtQP(f)](#function-evalatqp) | Evaluates a functor in global coordinates. [gradientAtQP(gf)](#function-gradientatqp) | Differentiate a grid function w.r.t. global coords [invokeAtQP(f,gf_0,gf_1...)](#function-invokeatqp) | Apply a functor to the evaluated grid functions [X()](#function-x), [X(comp)](#function-x) | Return (component of) the global coordinate constant | Any (constant) scalar value or FieldVector or FieldMatrix std::ref(variable) | A reference to an lvalue object +, -, *, / | Arithmetic expressions get(gf,i), get(gf) | Retrieve the i-th component of a vector expression: $ g(x)_i $ cmath Function | Descriptions -------------------------------------------|--------------------------------------------- max(gf_0,gf_1) | Maximum of two grid functions: $ \max\{g_0(x),g_1(x)\} $ min(gf_0,gf_1) | Minimum of two grid functions: $ \min\{g_0(x),g_1(x)\} $ abs_max(gf_0,gf_1) | Maximum of the absolute value of two grid functions: $ \max\{\vert g_0(x)\vert,\vert g_1(x)\vert\} $ abs_min(gf_0,gf_1) | Minimum of the absolute value of two grid functions: $ \min\{\vert g_0(x)\vert,\vert g_1(x)\vert\} $ clamp(gf,a,b) | A grid functions clamped btween two values a and b: $ \max\{a,\min\{b,g(x)\}\} $ abs(gf) | The absolute value of a grid functions: $ \vert g(x)\vert $ sqr(gf) | The square value of a grid functions: $ g(x)^2 $ pow(gf,p), pow
(gf) | A grid functions taken to the power of p: $ g(x)^p $ vector/matrix Function | Descriptions -------------------------------------------|--------------------------------------------- sum(gf) | The sum of the components of a vector-valued GridFunction: $ \sum_i g(x)_i $ unary_dot(gf) | The inner product of a vector-valued GridFunction with itself: $ \sum_i g(x)_i\cdot g(x)_i $ one_norm(gf) | The 1-norm of a vector-valued GridFunction: $ \sum_i \vert g(x)_i\vert $ two_norm(gf) | The 2-norm of a vector-valued GridFunction: $ \sqrt{\sum_i \vert g(x)_i\vert^2} $ p_norm
(gf) | The p-norm of a vector-valued GridFunction: $ (\sum_i \vert g(x)_i\vert^p)^{1/p} $ infty_norm(gf) | The infty-norm of a vector-valued GridFunction: $ \max_i \vert g(x)_i\vert $ trans(gf) | The transposed of a matrix-valued GridFunction: $ G(x)^T $ dot(gf_0,gf_1) | The inner product of two vector-valued GridFunctions: $ \sum_i g_0(x)_i\cdot g_1(x)_i $ distance(gf_0,gf_1) | The euclidean distance of two vector-valued GridFunctions: $ \sqrt{\sum_i \vert g_0(x)_i - g_1(x)_i\vert^2} $ outer(gf_0,gf_1) | The outer product of two vector-valued GridFunctions: $ G(x)_{ij} = g_0(x)_i\cdot g_1(x)_j $ ## function evalAtQP()  Müller, Felix committed Sep 29, 2020 155 Defined in header [](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/amdis/gridfunctions/AnalyticGridFunction.hpp)  Praetorius, Simon committed Apr 14, 2020 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  c++ template auto evalAtQP(Function const& f);  Creates a GridFunction that evaluates a functor in global coordinates. #### Arguments Function f : A callable object that can be called with global coordinates, i.e. typically FieldVector where dow is the world dimension. The range-type of the function determines the range-type of the expression. #### Requirements - Function models Concepts::CallableDomain ## function X()  Müller, Felix committed Sep 29, 2020 175 Defined in header [](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/amdis/gridfunctions/CoordsGridFunction.hpp)  Praetorius, Simon committed Apr 14, 2020 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192  c++ inline auto X(); // (1) inline auto X(int comp); // (2)  A GridFunction that evaluates to the global coordinates (1) or a component of the global coordinates (2). #### Arguments int comp : The component of the global coordinate vector. Should be 0 <= comp < dow where dow is the world dimension. ## function gradientAtQP()  Müller, Felix committed Sep 29, 2020 193 Defined in header [](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/amdis/gridfunctions/DerivativeGridFunction.hpp)  Praetorius, Simon committed Apr 14, 2020 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220  c++ template auto gradientAtQP(Expr const& expr);  Creates a GridFunction representing the gradient w.r.t. global coordinates of the wrapped expression. #### Arguments Expr expr : An expression that can be transformed into a grid function #### Requirements * The GridFunction of gf = makeGridFunction(expr, gridView)with some GridView models Concepts::GridFunction and its LocalFunction has a derivative, i.e. the expression derivative(localFunction(gf)) does not fail. #### Examples We assume there is a ProblemStat with the name prob. c++ gradientAtQP(prob.solution(_0)) gradientAtQP(X(0) + X(1) + prob.solution(_0))  ## function invokeAtQP()  Müller, Felix committed Sep 29, 2020 221 Defined in header [](https://gitlab.mn.tu-dresden.de/amdis/amdis-core/blob/master/amdis/gridfunctions/FunctorGridFunction.hpp)  Praetorius, Simon committed Apr 14, 2020 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254  c++ template auto invokeAtQP(Functor const& f, Exprs&&... g_i);  Creates a Gridfunction that applies a functor to the evaluated expressions. It creates a composition of GridFunctions g_i by applying a functor f locally, i.e. locally it is evaluated $ f(g_0(x), g_1(x), ...) $ #### Arguments Functor f : A Functor $f(...)$ that accepts as many arguments as there are grid functions Exprs g_i... : A number of expressions that can be transformed into grid functions $g_i$ #### Requirements * arity(f) == sizeof...(Exprs) * arity(g_i) == arity(g_j) for i != j * g_i models concept Concepts::GridFunction !!! note The composition of grid functions can be differentiated using [gradientAtQP()](#function-gradientatqp) if all the grid functions are differentiable and the functor is differentiable, i.e. there exist valid functions derivative(g_i) and partial(f, i). #### Examples c++ invokeAtQP([](Dune::FieldVector const& x) { return two_norm(x); }, X()); invokeAtQP([](double u, auto const& x) { return u + x[0]; }, 1.0, X()); invokeAtQP(Operation::Plus{}, X(0), X(1));