From: heltai Date: Sat, 18 Apr 2009 11:01:11 +0000 (+0000) Subject: Made integration clearer X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d8c613e941662210229269f57ad22ecec5b70970;p=dealii-svn.git Made integration clearer git-svn-id: https://svn.dealii.org/trunk@18646 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-34/step-34.cc b/deal.II/examples/step-34/step-34.cc index 484670cb00..aa2143ca2f 100644 --- a/deal.II/examples/step-34/step-34.cc +++ b/deal.II/examples/step-34/step-34.cc @@ -78,135 +78,40 @@ using namespace dealii; // the vector $R = // \mathbf{y}-\mathbf{x}$ is // different from zero. - // - // Whenever the integration is - // performed with the singularity - // inside the given cell, then a - // special quadrature formula is used - // that allows one to integrate - // arbitrary functions against a - // singular weight on the reference - // cell. There are two options when - // the integral is singular. One - // could take into account the - // singularity inside the quadrature - // formula as a weigthing function, - // or one could use a quadrature - // formula that is taylored to - // integrate singular objects, but - // where the actual weighting - // function is one. The use of the - // first method requires the user to - // provide "desingularized" single - // and double layer potentials which - // can then be integrated on the - // given cell. When the @p - // factor_out_singularity parameter - // is set to true, then the computed - // kernels do not conatain the - // singular factor, which is included - // in the quadrature formulas as a - // weighting function. This works - // best in two dimension, where the - // singular integrals are integrals - // along a segment of a logarithmic - // singularity. - // -//TODO: Can you elaborate in formulas? - // These integrals are somewhat - // delicate, because inserting a - // factor Jx in the variable of - // integration does not result only - // in a factor J appearing as a - // constant factor on the entire - // integral, but also on an - // additional integral to be added, - // that contains the logarithm of - // J. For this reason in two - // dimensions we opt for the - // desingularized kernel, and use the - // QGaussLogR quadrature formula, - // that takes care of integrating the - // correct weight for us. - // - // In the three dimensional case the - // singular integral is taken care of - // using the QGaussOneOverR - // quadrature formula. We could use - // the desingularized kernel here as - // well, but this would require us to - // be careful about the different - // scaling of $r$ in the reference - // cell and in real space. The - // quadrature formula uses as weight - // $1/r$ in local coordinates, while - // we need to integrate $1/R$ in real - // coordinates. A factor of $r/R$ has - // to be introduced in the quadrature - // formula. This can be done - // manually, or we simply calculate - // the standard kernels and then use - // a desingularized quadrature - // formula, i.e., one which is - // taylored for singular integrals, - // but whose weight is 1 instead of - // the singularity. - // - // Notice that the QGaussLog - // quadrature formula is made to - // integrate $f(x)\ln - // |\mathbf{x}-\mathbf{x}_0|$, but - // the kernel for two dimensional - // problems has the opposite - // sign. This is taken care of by - // switching the sign of the two - // dimensional desingularized kernel. - // - // The last argument to both - // functions is simply ignored in - // three dimensions. namespace LaplaceKernel { template - double single_layer(const Point &R, - const bool factor_out_2d_singularity = false) + double single_layer(const Point &R) { switch(dim) { case 2: - if (factor_out_2d_singularity == true) - return -1./(2*numbers::PI); - else - return (-std::log(R.norm()) / (2*numbers::PI) ); + return (-std::log(R.norm()) / (2*numbers::PI) ); case 3: - return (1./( R.norm()*4*numbers::PI ) ); + return (1./( R.norm()*4*numbers::PI ) ); default: - Assert(false, ExcInternalError()); - return 0.; + Assert(false, ExcInternalError()); + return 0.; } } template - Point double_layer(const Point &R, - const bool factor_out_2d_singularity = false) + Point double_layer(const Point &R) { switch(dim) { case 2: - if (factor_out_2d_singularity) - return Point(); - else - return R / (-2*numbers::PI * R.square()); + return R / ( -2*numbers::PI * R.square()); case 3: - return R / ( -4*numbers::PI * R.square()*R.norm() ); + return R / ( -4*numbers::PI * R.square()*R.norm() ); default: - Assert(false, ExcInternalError()); - return Point(); + Assert(false, ExcInternalError()); + return Point(); } } } @@ -452,10 +357,10 @@ class BEMProblem // well as a vector that will // hold the values of // $\alpha(\mathbf x)$ (the - // fraction of space visible from - // a point $\mathbf x$) at the - // support points of our shape - // functions. + // fraction of $\Omega$ visible + // from a point $\mathbf x$) at + // the support points of our + // shape functions. Vector phi; Vector alpha; @@ -1031,65 +936,189 @@ void BEMProblem::assemble_system() } } else { // Now we treat the more - // delicate case. If we are - // here, this means that the - // cell that runs on the $j$ - // index contains - // support_point[i]. In this - // case both the single and - // the double layer potential - // are singular, and they - // require special treatment, - // as explained in the - // introduction. + // delicate case. If we + // are here, this means + // that the cell that + // runs on the $j$ index + // contains + // support_point[i]. In + // this case both the + // single and the double + // layer potential are + // singular, and they + // require special + // treatment. // - // In the two dimensional - // case we perform the - // integration using a - // QGaussLogR quadrature - // formula, which is - // specifically designed to - // integrate logarithmic - // singularities on the unit - // interval, while in three - // dimensions we use the - // QGaussOneOverR class, - // which allows us to - // integrate 1/R - // singularities on the - // vertices of the reference - // element. Since we don't - // want to rebuild the two - // dimensional quadrature - // formula at each singular - // integration, we have built - // them outside the loop on - // the cells, and we only use - // a pointer to that - // quadrature here. + // Whenever the + // integration is + // performed with the + // singularity inside the + // given cell, then a + // special quadrature + // formula is used that + // allows one to + // integrate arbitrary + // functions against a + // singular weight on the + // reference cell. + // + // Notice that singular + // integration requires a + // careful selection of + // the quadrature + // rules. In particular + // the deal.II library + // provides quadrature + // rules which are + // taylored for + // logarithmic + // singularities + // (QGaussLog, + // QGaussLogR), as well + // as for 1/R + // singularities + // (QGaussOneOverR). + // + // Singular integration + // is typically obtained + // by constructing + // weighted quadrature + // formulas with singular + // weights, so that it is + // possible to write + // + // \f[ + // \int_K f(x) s(x) dx = \Sum_{i=1}^N w_i f(q_i) + // \f] + // + // where $s(x)$ is a + // given singularity. + // + // In all the finite + // element examples we + // have seen so far, the + // weight of the + // quadrature itself + // (namely, the function + // $s(x)$), was always + // constantly equal to 1. + // + // For singular + // integration, we have + // two choices: we can + // use the definition + // above, factoring out + // the singularity from + // the integrand (i.e., + // integrating $f(x)$ + // with the special + // quadrature rule), or + // we can ask the + // quadrature rule to + // "normalize" the + // weights $w_i$ with + // $s(q_i)$: + // + // \f[ + // \int_K f(x) s(x) dx = + // \int_K g(x) dx = \Sum_{i=1}^N \frac{w_i}{s(q_i)} g(q_i) + // \f] + // + // We use this second + // option, through the @p + // factor_out_singularity + // parameter of both + // QGaussLogR and + // QGaussOneOverR. + // + // These integrals are + // somewhat delicate, + // especially in two + // dimensions, due to the + // transformation from + // the real to the + // reference cell, where + // the variable of + // integration is scaled + // with the determinant + // of the transformation. + // + // In two dimensions this + // process does not + // result only in a + // factor appearing as a + // constant factor on the + // entire integral, but + // also on an additional + // integral alltogether + // that needs to be + // evaluated: + // + // \f[ + // \int_0^1 f(x)\ln(x/\alpha) dx = + // \int_0^1 f(x)\ln(x) dx - \int_0^1 f(x) \ln(\alpha) dx. + // \f] + // + // This process is taken care of by + // the constructor of the QGaussLogR + // class, which adds additional + // quadrature points and weights to + // take into consideration also the + // second part of the integral. + // + // A similar reasoning + // should be done in the + // three dimensional + // case, since the + // singular quadrature is + // taylored on the + // inverse of the radius + // $r$ in the reference + // cell, while our + // singular function + // lives in real space, + // however in the three + // dimensional case + // everything is simpler + // because the + // singularity scales + // linearly with the + // determinant of the + // transformation. This + // allows us to built the + // singular two + // dimensional quadrature + // rules once and for all + // outside the loop on + // the cells, using only + // a pointer where needed. // // Notice that in one - // dimensional integration - // this is not possible, - // since we need to know the - // scaling parameter for the - // quadrature, which is not - // known a priori. Here, the - // singular quadrature rule - // depends also on the size - // of the current cell. For - // this reason, it is - // necessary to create a new + // dimensional + // integration this is + // not possible, since we + // need to know the + // scaling parameter for + // the quadrature, which + // is not known a + // priori. Here, the + // quadrature rule itself + // depends also on the + // size of the current + // cell. For this reason, + // it is necessary to + // create a new // quadrature for each // singular // integration. Since we - // create it using the new - // operator of C++, we also - // need to destroy it using - // the dual of new: - // delete. This is done at - // the end, and only if dim - // == 2. + // create it using the + // new operator of C++, + // we also need to + // destroy it using the + // dual of new: + // delete. This is done + // at the end, and only + // if dim == 2. // // Putting all this into a // dimension independent @@ -1124,7 +1153,7 @@ void BEMProblem::assemble_system() dynamic_cast*>( new QGaussLogR<1>(singular_quadrature_order, Point<1>((double)singular_index), - 1./cell->measure())) + 1./cell->measure(), true)) : (dim == 3 ? @@ -1158,15 +1187,15 @@ void BEMProblem::assemble_system() normal_wind += (singular_cell_wind[q](d)* singular_normals[q][d]); - system_rhs(i) += ( LaplaceKernel::single_layer(R, is_singular) * + system_rhs(i) += ( LaplaceKernel::single_layer(R) * normal_wind * fe_v_singular.JxW(q) ); for(unsigned int j=0; j