void RefMap::calc_second_ref_map(int order)
{
assert(quad_2d != NULL);
int i, j, np = quad_2d->get_num_points(order);
AUTOLA_OR(double3x2, k, np);
memset(k, 0, k.size);
ref_map_pss.force_transform(sub_idx, ctm);
for (i = 0; i < nc; i++)
{
double *dxy, *dxx, *dyy;
ref_map_pss.set_active_shape(indices[i]);
ref_map_pss.set_quad_order(order, H2D_FN_ALL);
dxx = ref_map_pss.get_dxx_values();
dyy = ref_map_pss.get_dyy_values();
dxy = ref_map_pss.get_dxy_values();
for (j = 0; j < np; j++)
{
k[j][0][0] += coeffs[i][0] * dxx[j];
k[j][0][1] += coeffs[i][1] * dxx[j];
k[j][1][0] += coeffs[i][0] * dxy[j];
k[j][1][1] += coeffs[i][1] * dxy[j];
k[j][2][0] += coeffs[i][0] * dyy[j];
k[j][2][1] += coeffs[i][1] * dyy[j];
}
}
double3x2* mm = cur_node->second_ref_map[order] = new double3x2[np];
double2x2* m = get_inv_ref_map(order);
for (j = 0; j < np; j++)
{
double a, b;
// coefficients in second derivative with respect to xx
a = sqr(m[j][0][0])*k[j][0][0] + 2*m[j][0][0]*m[j][0][1]*k[j][1][0] + sqr(m[j][0][1])*k[j][2][0];
b = sqr(m[j][0][0])*k[j][0][1] + 2*m[j][0][0]*m[j][0][1]*k[j][1][1] + sqr(m[j][0][1])*k[j][2][1];
mm[j][0][0] = -(a * m[j][0][0] + b * m[j][1][0]); // du/dx
mm[j][0][1] = -(a * m[j][0][1] + b * m[j][1][1]); // du/dy
// coefficients in second derivative with respect to xy
a = m[j][0][0]*m[j][1][0]*k[j][0][0] + (m[j][0][1]*m[j][1][0] + m[j][0][0]*m[j][1][1])*k[j][1][0] + m[j][0][1]*m[j][1][1]*k[j][2][0];
b = m[j][0][0]*m[j][1][0]*k[j][0][1] + (m[j][0][1]*m[j][1][0] + m[j][0][0]*m[j][1][1])*k[j][1][1] + m[j][0][1]*m[j][1][1]*k[j][2][1];
mm[j][1][0] = -(a * m[j][0][0] + b * m[j][1][0]); // du/dx
mm[j][1][1] = -(a * m[j][0][1] + b * m[j][1][1]); // du/dy
// coefficients in second derivative with respect to yy
a = sqr(m[j][1][0])*k[j][0][0] + 2*m[j][1][0]*m[j][1][1]*k[j][1][0] + sqr(m[j][1][1])*k[j][2][0];
b = sqr(m[j][1][0])*k[j][0][1] + 2*m[j][1][0]*m[j][1][1]*k[j][1][1] + sqr(m[j][1][1])*k[j][2][1];
mm[j][2][0] = -(a * m[j][0][0] + b * m[j][1][0]); // du/dx
mm[j][2][1] = -(a * m[j][0][1] + b * m[j][1][1]); // du/dy
}
}
void RefMap::calc_phys_y(int order)
{
// transform all y coordinates of the integration points
int i, j, np = quad_2d->get_num_points(order);
double* y = cur_node->phys_y[order] = new double[np];
memset(y, 0, np * sizeof(double));
ref_map_pss.force_transform(sub_idx, ctm);
for (i = 0; i < nc; i++)
{
ref_map_pss.set_active_shape(indices[i]);
ref_map_pss.set_quad_order(order);
double* fn = ref_map_pss.get_fn_values();
for (j = 0; j < np; j++)
y[j] += coeffs[i][1] * fn[j];
}
}
void RefMap::calc_inv_ref_map(int order)
{
assert(quad_2d != NULL);
int i, j, np = quad_2d->get_num_points(order);
// construct jacobi matrices of the direct reference map for all integration points
AUTOLA_OR(double2x2, m, np);
memset(m, 0, m.size);
ref_map_pss.force_transform(sub_idx, ctm);
for (i = 0; i < nc; i++)
{
double *dx, *dy;
ref_map_pss.set_active_shape(indices[i]);
ref_map_pss.set_quad_order(order);
ref_map_pss.get_dx_dy_values(dx, dy);
for (j = 0; j < np; j++)
{
m[j][0][0] += coeffs[i][0] * dx[j];
m[j][0][1] += coeffs[i][0] * dy[j];
m[j][1][0] += coeffs[i][1] * dx[j];
m[j][1][1] += coeffs[i][1] * dy[j];
}
}
// calculate the jacobian and inverted matrix
double trj = get_transform_jacobian();
double2x2* irm = cur_node->inv_ref_map[order] = new double2x2[np];
double* jac = cur_node->jacobian[order] = new double[np];
for (i = 0; i < np; i++)
{
jac[i] = (m[i][0][0] * m[i][1][1] - m[i][0][1] * m[i][1][0]);
double ij = 1.0 / jac[i];
error_if(!finite(ij), "1/jac[%d] is infinity when calculating inv. ref. map for order %d (jac=%g)", i, order);
assert_msg(ij == ij, "1/jac[%d] is NaN when calculating inv. ref. map for order %d (jac=%g)", i, order);
// invert and transpose the matrix
irm[i][0][0] = m[i][1][1] * ij;
irm[i][0][1] = -m[i][1][0] * ij;
irm[i][1][0] = -m[i][0][1] * ij;
irm[i][1][1] = m[i][0][0] * ij;
jac[i] *= trj;
}
}
void RefMap::calc_tangent(int edge, int eo)
{
int i, j;
int np = quad_2d->get_num_points(eo);
double3* tan = cur_node->tan[edge] = new double3[np];
int a = edge, b = element->next_vert(edge);
if (!element->is_curved())
{
// straight edges: the tangent at each point is just the edge length
tan[0][0] = element->vn[b]->x - element->vn[a]->x;
tan[0][1] = element->vn[b]->y - element->vn[a]->y;
tan[0][2] = sqrt(sqr(tan[0][0]) + sqr(tan[0][1]));
double inorm = 1.0 / tan[0][2];
tan[0][0] *= inorm;
tan[0][1] *= inorm;
tan[0][2] *= (edge == 0 || edge == 2) ? ctm->m[0] : ctm->m[1];
for (i = 1; i < np; i++)
memcpy(tan+i, tan, sizeof(double3));
}
else
{
// construct jacobi matrices of the direct reference map at integration points along the edge
static double2x2 m[15];
assert(np <= 15);
memset(m, 0, np*sizeof(double2x2));
ref_map_pss.force_transform(sub_idx, ctm);
for (i = 0; i < nc; i++)
{
double *dx, *dy;
ref_map_pss.set_active_shape(indices[i]);
ref_map_pss.set_quad_order(eo);
ref_map_pss.get_dx_dy_values(dx, dy);
for (j = 0; j < np; j++)
{
m[j][0][0] += coeffs[i][0] * dx[j];
m[j][0][1] += coeffs[i][0] * dy[j];
m[j][1][0] += coeffs[i][1] * dx[j];
m[j][1][1] += coeffs[i][1] * dy[j];
}
}
// multiply them by the vector of the reference edge
double2* v1 = ref_map_shapeset.get_ref_vertex(a);
double2* v2 = ref_map_shapeset.get_ref_vertex(b);
double ex = (*v2)[0] - (*v1)[0];
double ey = (*v2)[1] - (*v1)[1];
for (i = 0; i < np; i++)
{
double3& t = tan[i];
t[0] = m[i][0][0]*ex + m[i][0][1]*ey;
t[1] = m[i][1][0]*ex + m[i][1][1]*ey;
t[2] = sqrt(sqr(t[0]) + sqr(t[1]));
double inorm = 1.0 / t[2];
t[0] *= inorm;
t[1] *= inorm;
t[2] *= (edge == 0 || edge == 2) ? ctm->m[0] : ctm->m[1];
}
}
}