public static void assemble_poisson(int node_num, double[] node_xy,
int element_num, int[] element_node, int quad_num, int ib, ref double[] a,
ref double[] f, Func <int, double[], double[]> rhs, Func <int, double[], double[]> h_coef,
Func <int, double[], double[]> k_coef)
//****************************************************************************80
//
// Purpose:
//
// ASSEMBLE_POISSON assembles the system for the Poisson equation.
//
// Discussion:
//
// The matrix is known to be banded. A special matrix storage format
// is used to reduce the space required. Details of this format are
// discussed in the routine DGB_FA.
//
// Note that a 3 point quadrature rule, which is sometimes used to
// assemble the matrix and right hand side, is just barely accurate
// enough for simple problems. If you want better results, you
// should use a quadrature rule that is more accurate.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 06 December 2010
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int NODE_NUM, the number of nodes.
//
// Input, double NODE_XY[2*NODE_NUM], the coordinates of nodes.
//
// Input, int ELEMENT_NUM, the number of elements.
//
// Input, int ELEMENT_NODE[3*ELEMENT_NUM];
// ELEMENT_NODE(I,J) is the global index of local node I in element J.
//
// Input, int QUAD_NUM, the number of quadrature points used in assembly.
//
// Input, int IB, the half-bandwidth of the matrix.
//
// Output, double A(3*IB+1,NODE_NUM), the NODE_NUM by NODE_NUM
// coefficient matrix, stored in a compressed format.
//
// Output, double F(NODE_NUM), the right hand side.
//
// Local parameters:
//
// Local, double BI, DBIDX, DBIDY, the value of some basis function
// and its first derivatives at a quadrature point.
//
// Local, double BJ, DBJDX, DBJDY, the value of another basis
// function and its first derivatives at a quadrature point.
//
{
double bi = 0;
double bj = 0;
double dbidx = 0;
double dbidy = 0;
double dbjdx = 0;
double dbjdy = 0;
int element;
int i;
int node;
double[] p = new double[2];
double[] t3 = new double[2 * 3];
double[] phys_h = new double[quad_num];
double[] phys_k = new double[quad_num];
double[] phys_rhs = new double[quad_num];
double[] phys_xy = new double[2 * quad_num];
double[] quad_w = new double[quad_num];
double[] quad_xy = new double[2 * quad_num];
double[] w = new double[quad_num];
//
// Initialize the arrays to zero.
//
for (node = 0; node < node_num; node++)
{
f[node] = 0.0;
}
for (node = 0; node < node_num; node++)
{
for (i = 0; i < 3 * ib + 1; i++)
{
a[i + node * (3 * ib + 1)] = 0.0;
}
}
//
// Get the quadrature weights and nodes.
//
QuadratureRule.quad_rule(quad_num, ref quad_w, ref quad_xy);
//
// Add up all quantities associated with the ELEMENT-th element.
//
for (element = 0; element < element_num; element++)
{
//
// Make a copy of the element.
//
int j;
for (j = 0; j < 3; j++)
{
for (i = 0; i < 2; i++)
{
t3[i + j * 2] = node_xy[i + (element_node[j + element * 3] - 1) * 2];
}
}
//
// Map the quadrature points QUAD_XY to points PHYS_XY in the physical element.
//
Reference.reference_to_physical_t3(t3, quad_num, quad_xy, ref phys_xy);
double area = Math.Abs(typeMethods.triangle_area_2d(t3));
int quad;
for (quad = 0; quad < quad_num; quad++)
{
w[quad] = quad_w[quad] * area;
}
phys_rhs = rhs(quad_num, phys_xy);
phys_h = h_coef(quad_num, phys_xy);
phys_k = k_coef(quad_num, phys_xy);
//
// Consider the QUAD-th quadrature point.
//
for (quad = 0; quad < quad_num; quad++)
{
p[0] = phys_xy[0 + quad * 2];
p[1] = phys_xy[1 + quad * 2];
//
// Consider the TEST-th test function.
//
// We generate an integral for every node associated with an unknown.
// But if a node is associated with a boundary condition, we do nothing.
//
int test;
for (test = 1; test <= 3; test++)
{
i = element_node[test - 1 + element * 3];
Basis11.basis_one_t3(t3, test, p, ref bi, ref dbidx, ref dbidy);
f[i - 1] += w[quad] * phys_rhs[quad] * bi;
//
// Consider the BASIS-th basis function, which is used to form the
// value of the solution function.
//
int basis;
for (basis = 1; basis <= 3; basis++)
{
j = element_node[basis - 1 + element * 3];
Basis11.basis_one_t3(t3, basis, p, ref bj, ref dbjdx, ref dbjdy);
a[i - j + 2 * ib + (j - 1) * (3 * ib + 1)] += w[quad] * (
phys_h[quad] *
(dbidx * dbjdx + dbidy * dbjdy)
+ phys_k[quad] * bj * bi);
}
}
}
}
}
public static void assemble_poisson_dsp(int node_num, double[] node_xy,
int element_num, int[] element_node, int quad_num, int nz_num, int[] ia,
int[] ja, ref double[] a, ref double[] f, Func <int, double[], double[]> rhs, Func <int, double[], double[]> h_coef, Func <int, double[], double[]> k_coef)
//****************************************************************************80
//
// Purpose:
//
// ASSEMBLE_POISSON_DSP assembles the system for the Poisson equation.
//
// Discussion:
//
// The matrix is sparse, and stored in the DSP or "sparse triple" format.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 13 July 2007
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int NODE_NUM, the number of nodes.
//
// Input, double NODE_XY[2*NODE_NUM], the
// coordinates of nodes.
//
// Input, int ELEMENT_NUM, the number of triangles.
//
// Input, int ELEMENT_NODE[3*ELEMENT_NUM];
// ELEMENT_NODE(I,J) is the global index of local node I in triangle J.
//
// Input, int QUAD_NUM, the number of quadrature points used in assembly.
//
// Input, int NZ_NUM, the number of nonzero entries.
//
// Input, int IA[NZ_NUM], JA[NZ_NUM], the row and column
// indices of the nonzero entries.
//
// Output, double A[NZ_NUM], the nonzero entries of the matrix.
//
// Output, double F(NODE_NUM), the right hand side.
//
// Local parameters:
//
// Local, double BI, DBIDX, DBIDY, the value of some basis function
// and its first derivatives at a quadrature point.
//
// Local, double BJ, DBJDX, DBJDY, the value of another basis
// function and its first derivatives at a quadrature point.
//
{
double bi = 0;
double bj = 0;
double dbidx = 0;
double dbidy = 0;
double dbjdx = 0;
double dbjdy = 0;
int element;
int node;
int nz;
double[] p = new double[2];
double[] t3 = new double[2 * 3];
double[] phys_xy = new double[2 * quad_num];
double[] quad_w = new double[quad_num];
double[] quad_xy = new double[2 * quad_num];
double[] w = new double[quad_num];
//
// Initialize the arrays to zero.
//
for (node = 0; node < node_num; node++)
{
f[node] = 0.0;
}
for (nz = 0; nz < nz_num; nz++)
{
a[nz] = 0.0;
}
//
// Get the quadrature weights and nodes.
//
QuadratureRule.quad_rule(quad_num, ref quad_w, ref quad_xy);
//
// Add up all quantities associated with the ELEMENT-th element.
//
for (element = 0; element < element_num; element++)
{
//
// Make a copy of the element.
//
int j;
int i;
for (j = 0; j < 3; j++)
{
for (i = 0; i < 2; i++)
{
t3[i + j * 2] = node_xy[i + (element_node[j + element * 3] - 1) * 2];
}
}
//
// Map the quadrature points QUAD_XY to points XY in the physical element.
//
Reference.reference_to_physical_t3(t3, quad_num, quad_xy, ref phys_xy);
double area = Math.Abs(typeMethods.triangle_area_2d(t3));
int quad;
for (quad = 0; quad < quad_num; quad++)
{
w[quad] = quad_w[quad] * area;
}
double[] phys_rhs = rhs(quad_num, phys_xy);
double[] phys_h = h_coef(quad_num, phys_xy);
double[] phys_k = k_coef(quad_num, phys_xy);
//
// Consider the QUAD-th quadrature point.
//
for (quad = 0; quad < quad_num; quad++)
{
p[0] = phys_xy[0 + quad * 2];
p[1] = phys_xy[1 + quad * 2];
//
// Consider the TEST-th test function.
//
// We generate an integral for every node associated with an unknown.
// But if a node is associated with a boundary condition, we do nothing.
//
int test;
for (test = 1; test <= 3; test++)
{
i = element_node[test - 1 + element * 3];
Basis11.basis_one_t3(t3, test, p, ref bi, ref dbidx, ref dbidy);
f[i - 1] += w[quad] * phys_rhs[quad] * bi;
//
// Consider the BASIS-th basis function, which is used to form the
// value of the solution function.
//
int basis;
for (basis = 1; basis <= 3; basis++)
{
j = element_node[basis - 1 + element * 3];
Basis11.basis_one_t3(t3, basis, p, ref bj, ref dbjdx, ref dbjdy);
int k = dsp_ij_to_k(nz_num, ia, ja, i, j);
a[k - 1] += w[quad] * (
phys_h[quad] * (dbidx * dbjdx + dbidy * dbjdy)
+ phys_k[quad] * bj * bi);
}
}
}
}
}
public static double[] residual_poisson(int node_num, double[] node_xy, int[] node_condition,
int element_num, int[] element_node, int quad_num, int ib, double[] a,
double[] f, double[] node_u, Func <int, double[], double[]> rhs, Func <int, double[], double[]> h_coef,
Func <int, double[], double[]> k_coef, Func <int, double[], double[]> dirichlet_condition)
//****************************************************************************80
//
// Purpose:
//
// RESIDUAL_POISSON evaluates the residual for the Poisson equation.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 14 November 2006
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int NODE_NUM, the number of nodes.
//
// Input, double NODE_XY[2*NODE_NUM], the
// coordinates of nodes.
//
// Input, int NODE_CONDITION[NODE_NUM], reports the condition
// used to set the unknown associated with the node.
// 0, unknown.
// 1, finite element equation.
// 2, Dirichlet condition;
// 3, Neumann condition.
//
// Input, int ELEMENT_NUM, the number of elements.
//
// Input, int ELEMENT_NODE[3*ELEMENT_NUM];
// ELEMENT_NODE(I,J) is the global index of local node I in element J.
//
// Input, int QUAD_NUM, the number of quadrature points used in assembly.
//
// Input, int IB, the half-bandwidth of the matrix.
//
// Workspace, double A[(3*IB+1)*NODE_NUM], the NODE_NUM by NODE_NUM
// coefficient matrix, stored in a compressed format.
//
// Workspace, double F[NODE_NUM], the right hand side.
//
// Input, double NODE_U[NODE_NUM], the value of the solution
// at each node.
//
// Output, double NODE_R[NODE_NUM], the finite element
// residual at each node.
//
// Local parameters:
//
// Local, double BI, DBIDX, DBIDY, the value of some basis function
// and its first derivatives at a quadrature point.
//
// Local, double BJ, DBJDX, DBJDY, the value of another basis
// function and its first derivatives at a quadrature point.
//
{
double bi = 0;
double bj = 0;
double dbidx = 0;
double dbidy = 0;
double dbjdx = 0;
double dbjdy = 0;
int element;
int i;
int node;
double[] p = new double[2];
double[] t3 = new double[2 * 3];
double[] phys_h = new double[quad_num];
double[] phys_k = new double[quad_num];
double[] phys_rhs = new double[quad_num];
double[] phys_xy = new double[2 * quad_num];
double[] quad_w = new double[quad_num];
double[] quad_xy = new double[2 * quad_num];
double[] w = new double[quad_num];
//
// Initialize the arrays to zero.
//
for (node = 0; node < node_num; node++)
{
f[node] = 0.0;
}
for (node = 0; node < node_num; node++)
{
for (i = 0; i < 3 * ib + 1; i++)
{
a[i + node * (3 * ib + 1)] = 0.0;
}
}
//
// Get the quadrature weights and nodes.
//
QuadratureRule.quad_rule(quad_num, ref quad_w, ref quad_xy);
//
// The actual values of A and F are determined by summing up
// contributions from all the elements.
//
for (element = 0; element < element_num; element++)
{
//
// Make a copy of the element.
//
int j;
for (j = 0; j < 3; j++)
{
for (i = 0; i < 2; i++)
{
t3[i + j * 2] = node_xy[i + (element_node[j + element * 3] - 1) * 2];
}
}
//
// Map the quadrature points QUAD_XY to points XY in the physical element.
//
Reference.reference_to_physical_t3(t3, quad_num, quad_xy, ref phys_xy);
double area = Math.Abs(typeMethods.triangle_area_2d(t3));
int quad;
for (quad = 0; quad < quad_num; quad++)
{
w[quad] = quad_w[quad] * area;
}
phys_rhs = rhs(quad_num, phys_xy);
phys_h = h_coef(quad_num, phys_xy);
phys_k = k_coef(quad_num, phys_xy);
//
// Consider a quadrature point QUAD, with coordinates (X,Y).
//
for (quad = 0; quad < quad_num; quad++)
{
p[0] = phys_xy[0 + quad * 2];
p[1] = phys_xy[1 + quad * 2];
//
// Consider one of the basis functions, which will play the
// role of test function in the integral.
//
// We generate an integral for every node associated with an unknown.
// But if a node is associated with a boundary condition, we do nothing.
//
int test;
for (test = 1; test <= 3; test++)
{
i = element_node[test - 1 + element * 3];
Basis11.basis_one_t3(t3, test, p, ref bi, ref dbidx, ref dbidy);
f[i - 1] += w[quad] * phys_rhs[quad] * bi;
//
// Consider another basis function, which is used to form the
// value of the solution function.
//
int basis;
for (basis = 1; basis <= 3; basis++)
{
j = element_node[basis - 1 + element * 3];
Basis11.basis_one_t3(t3, basis, p, ref bj, ref dbjdx, ref dbjdy);
a[i - j + 2 * ib + (j - 1) * (3 * ib + 1)] += w[quad]
* (phys_h[quad] *
(dbidx * dbjdx + dbidy * dbjdy)
+ phys_k[quad] * bj * bi);
}
}
}
}
//
// Apply boundary conditions.
//
dirichlet_apply(node_num, node_xy, node_condition, ib, ref a, ref f, dirichlet_condition);
//
// Compute A*U.
//
double[] node_r = dgb_mxv(node_num, node_num, ib, ib, a, node_u);
//
// Set RES = A * U - F.
//
for (node = 0; node < node_num; node++)
{
node_r[node] -= f[node];
}
return(node_r);
}
