Załącznik 3
Kod MATLABa obliczający problem transportu ciepła na obszarze w kształcie litery L
(zob. rozdział Implementacja w MATLABie problemu transportu ciepła )
1;
% Build cartesian product of specified vectors.
% Vector orientation is arbitrary.
%
% Order: first component changes fastest
%
% a1, a2, ... - sequence of n vectors
%
% returns - array of n-columns containing all the combinations of values in aj
function c = cartesian(varargin)
n = nargin;
[F{1:n}] = ndgrid(varargin{:});
for i = n:-1:1
c(i,:) = F{i}(:);
end
end
% Create a row vector of size n filled with val
function r = row_of(val, n)
r = val * ones(1, n);
end
% Index conventions
%------------------
%
% DoFs - zero-based
% Elements - zero-based
% Knot elements - zero-based
% Linear indices - one-based
% Create an one-dimensional basis object from specified data.
% Performs some simple input validation.
%
% For a standard, clamped B-spline basis first and last elements of the knot vector
% should be repeated (p+1) times.
%
% p - polynomial order
% points - increasing sequence of values defining the mesh
% knot - knot vector containing integer indices of mesh points (starting from 0)
%
% returns - structure describing the basis
function b = basis1d(p, points, knot)
validateattributes(points, {}, {'increasing'});
validateattributes(knot, {}, {'nondecreasing'});
assert(max(knot) == length(points) - 1, sprintf('Invalid knot index: %d, points: %d)', max(knot), length(points)));
b.p = p;
b.points = points;
b.knot = knot;
endfunction
% Number of basis functions (DoFs) in the 1D basis
function n = number_of_dofs(b)
n = length(b.knot) - b.p - 1;
endfunction
% Number of elements the domain is subdivided into
function n = number_of_elements(b)
n = length(b.points) - 1;
endfunction
% Domain point corresponding to i-th element of the knot vector
function x = knot_point(b, i)
x = b.points(b.knot(i) + 1);
endfunction
% Row vector containing indices of all the DoFs
function idx = dofs1d(b)
n = number_of_dofs(b);
idx = 0 : n-1;
endfunction
% Enumerate degrees of freedom in a tensor product of 1D bases
%
% b1, b2, ... - sequence of n 1D bases
%
% returns - array of indices (n-columns) of basis functions
function idx = dofs(varargin)
if (nargin == 1)
idx = dofs1d(varargin{:});
else
ranges = cellfun(@(b) dofs1d(b), varargin, 'UniformOutput', false);
idx = cartesian(ranges{:});
endif
endfunction
% Row vector containing indices of all the elements
function idx = elements1d(b)
n = number_of_elements(b);
idx = 0 : n-1;
endfunction
% Enumerate element indices for a tensor product of 1D bases
%
% b1, b2, ... - sequence of n 1D bases
%
% returns - array of indices (n-columns) of element indices
function idx = elements(varargin)
if (nargin == 1)
idx = elements1d(varargin{:});
else
ranges = cellfun(@(b) elements1d(b), varargin, 'UniformOutput', false);
idx = cartesian(ranges{:});
endif
endfunction
% Index of the first DoF that is non-zero over the specified element
function idx = first_dof_on_element(e, b)
idx = lookup(b.knot, e) - b.p - 1;
endfunction
% Row vector containing indices of DoFs that are non-zero over the specified element
%
% e - element index (scalar)
% b - 1D basis
function idx = dofs_on_element1d(e, b)
a = first_dof_on_element(e, b);
idx = a : a + b.p;
endfunction
% Row vector containing indices (columns) of DoFs that are non-zero over the specified element
%
% e - element index (pair)
% bx, by - 1D bases
function idx = dofs_on_element2d(e, bx, by)
rx = dofs_on_element1d(e(1), bx);
ry = dofs_on_element1d(e(2), by);
idx = cartesian(rx, ry);
endfunction
% Determine which edges of the element lie on the domain boundary
%
% e - element index (pair)
% bx, by - 1D bases
%
% returns - array of 4 boolean values (0 or 1), 1 meaning the edge is part of domain boundary
% Order of the edges:
% 1 - left
% 2 - right
% 3 - top
% 4 - bottom
function s = boundary_edges(e, bx, by)
nx = number_of_elements(bx);
ny = number_of_elements(by);
s = [e(1) == 0, ... % left
e(1) == nx - 1, ... % right
e(2) == ny - 1, ... % top
e(2) == 0]; % bottom
endfunction
% Compute 1-based, linear index of tensor product DoF.
% Column-major order - first index component changes fastest.
%
% dof - n-tuple index
% b1, b2,, ... - sequence of n 1D bases
%
% returns - linearized scalar index
function idx = linear_index(dof, varargin)
n = length(varargin);
idx = dof(n);
for i = n-1 : -1 : 1
ni = number_of_dofs(varargin{i});
idx = dof(i) + idx * ni;
endfor
idx += 1;
endfunction
% Assuming clamped B-spline basis, compute the polynomial order based on the knot
function p = degree_from_knot(knot)
p = find(knot > 0, 1) - 2;
endfunction
% Spline evaluation functions are based on:
%
% The NURBS Book, L. Piegl, W. Tiller, Springer 1995
% Find index i such that x lies between points corresponding to knot(i) and knot(i+1)
function span = find_span(x, b)
low = b.p + 1;
high = number_of_dofs(b) + 1;
if (x >= knot_point(b, high))
span = high - 1;
elseif (x <= knot_point(b, low))
span = low;
else
span = floor((low + high) / 2);
while (x < knot_point(b, span) || x >= knot_point(b, span + 1))
if (x < knot_point(b, span))
high = span;
else
low = span;
endif
span = floor((low + high) / 2);
endwhile
endif
endfunction
% Compute values at point x of (p+1) basis functions that are nonzero over the element
% corresponding to specified span.
%
% span - span containing x, as computed by function find_span
% x - point of evaluation
% b - basis
%
% returns - vector of size (p+1)
function out = evaluate_bspline_basis(span, x, b)
p = b.p;
out = zeros(p + 1, 1);
left = zeros(p, 1);
right = zeros(p, 1);
out(1) = 1;
for j = 1:p
left(j) = x - knot_point(b, span + 1 - j);
right(j) = knot_point(b, span + j) - x;
saved = 0;
for r = 1:j
tmp = out(r) / (right(r) + left(j - r + 1));
out(r) = saved + right(r) * tmp;
saved = left(j - r + 1) * tmp;
endfor
out(j + 1) = saved;
endfor
endfunction
% Compute values and derivatives of order up to der at point x of (p+1) basis functions
% that are nonzero over the element corresponding to specified span.
%
% span - span containing x, as computed by function find_span
% x - point of evaluation
% b - basis
%
% returns - array of size (p+1) x (der + 1) containing values and derivatives
function out = evaluate_bspline_basis_ders(span, x, b, der)
p = b.p;
out = zeros(p + 1, der + 1);
left = zeros(p, 1);
right = zeros(p, 1);
ndu = zeros(p + 1, p + 1);
a = zeros(2, p + 1);
ndu(1, 1) = 1;
for j = 1:p
left(j) = x - knot_point(b, span + 1 - j);
right(j) = knot_point(b, span + j) - x;
saved = 0;
for r = 1:j
ndu(j + 1, r) = right(r) + left(j - r + 1);
tmp = ndu(r, j) / ndu(j + 1, r);
ndu(r, j + 1) = saved + right(r) * tmp;
saved = left(j - r + 1) * tmp;
endfor
ndu(j + 1, j + 1) = saved;
endfor
out(:, 1) = ndu(:, p + 1);
for r = 0:p
s1 = 1;
s2 = 2;
a(1, 1) = 1;
for k = 1:der
d = 0;
rk = r - k;
pk = p - k;
if (r >= k)
a(s2, 1) = a(s1, 1) / ndu(pk + 2, rk + 1);
d = a(s2, 1) * ndu(rk + 1, pk + 1);
endif
j1 = max(-rk, 1);
if (r - 1 <= pk)
j2 = k - 1;
else
j2 = p - r;
endif
for j = j1:j2
a(s2, j + 1) = (a(s1, j + 1) - a(s1, j)) / ndu(pk + 2, rk + j + 1);
d = d + a(s2, j + 1) * ndu(rk + j + 1, pk + 1);
endfor
if (r <= pk)
a(s2, k + 1) = -a(s1, k) / ndu(pk + 2, r + 1);
d = d + a(s2, k + 1) * ndu(r + 1, pk + 1);
endif
out(r + 1, k + 1) = d;
t = s1;
s1 = s2;
s2 = t;
endfor
endfor
r = p;
for k = 1:der
for j = 1:p+1
out(j, k + 1) = out(j, k + 1) * r;
endfor
r = r * (p - k);
endfor
endfunction
% Evaluate combination of 2D B-splines at point x
function val = evaluate2d(u, x, bx, by)
sx = find_span(x(1), bx);
sy = find_span(x(2), by);
valsx = evaluate_bspline_basis(sx, x(1), bx);
valsy = evaluate_bspline_basis(sy, x(2), by);
offx = sx - bx.p;
offy = sy - by.p;
val = 0;
for i = 0:bx.p
for j = 0:by.p
val = val + u(offx + i, offy + j) * valsx(i + 1) * valsy(j + 1);
endfor
endfor
endfunction
% Returns a structure containing information about 1D basis functions that can be non-zero at x,
% with the following fields:
% offset - difference between global DoF numbers and indices into vals array
% vals - array of size (p+1) x (der + 1) containing values and derivatives of basis functions at x
function data = eval_local_basis(x, b, ders)
span = find_span(x, b);
first = span - b.p - 1;
data.offset = first - 1;
data.vals = evaluate_bspline_basis_ders(span, x, b, ders);
endfunction
% Compute value and derivative of specified 1D basis function, given data computed
% by function eval_local_basis
function [v, dv] = eval_dof1d(dof, data, b)
v = data.vals(dof - data.offset, 1);
dv = data.vals(dof - data.offset, 2);
endfunction
% Compute value and gradient of specified 2D basis function, given data computed
% by function eval_local_basis
function [v, dv] = eval_dof2d(dof, datax, datay, bx, by)
[a, da] = eval_dof1d(dof(1), datax, bx);
[b, db] = eval_dof1d(dof(2), datay, by);
v = a * b;
dv = [da * b, a * db];
endfunction
% Creates a wrapper function that takes 2D basis function index as argument and returns
% its value and gradient
function f = basis_evaluator2d(x, bx, by, ders)
datax = eval_local_basis(x(1), bx, 1);
datay = eval_local_basis(x(2), by, 1);
f = @(i) eval_dof2d(i, datax, datay, bx, by);
endfunction
% Value of 1D element mapping jacobian (size of the element)
function a = jacobian1d(e, b)
a = b.points(e + 2) - b.points(e + 1);
endfunction
% Value of 2D element mapping jacobian (size of the element)
function a = jacobian2d(e, bx, by)
a = jacobian1d(e(1), bx) * jacobian1d(e(2), by);
endfunction
% Row vector of points of the k-point Gaussian quadrature on [a, b]
function xs = quad_points(a, b, k)
% Affine mapping [-1, 1] -> [a, b]
map = @(x) 0.5 * (a * (1 - x) + b * (x + 1));
switch (k)
case 1
xs = [0];
case 2
xs = [-0.5773502691896257645, ...
0.5773502691896257645];
case 3
xs = [-0.7745966692414833770, ...
0, ...
0.7745966692414833770];
case 4
xs = [-0.8611363115940525752, ...
-0.3399810435848562648, ...
0.3399810435848562648, ...
0.8611363115940525752];
case 5
xs = [-0.9061798459386639928, ...
-0.5384693101056830910, ...
0, ...
0.5384693101056830910, ...
0.9061798459386639928];
endswitch
xs = map(xs);
endfunction
% Row vector of weights of the k-point Gaussian quadrature on [a, b]
function ws = quad_weights(k)
switch (k)
case 1
ws = [2];
case 2
ws = [1, 1];
case 3
ws = [0.55555555555555555556, ...
0.88888888888888888889, ...
0.55555555555555555556];
case 4
ws = [0.34785484513745385737, ...
0.65214515486254614263, ...
0.65214515486254614263, ...
0.34785484513745385737];
case 5
ws = [0.23692688505618908751, ...
0.47862867049936646804, ...
0.56888888888888888889, ...
0.47862867049936646804, ...
0.23692688505618908751]
endswitch
% Gaussian quadrature is defined on [-1, 1], we use [0, 1]
ws = ws / 2;
endfunction
% Create array of structures containing quadrature data for integrating over 1D element
%
% e - element index
% k - quadrature order
% b - 1D basis
%
% returns - array of k structures with fields
% x - point
% w - weight
function qs = quad_data1d(e, k, b)
xs = quad_points(b.points(e(1) + 1), b.points(e(1) + 2), k);
ws = quad_weights(k);
for i = 1:k
qs(i).x = xs(i);
qs(i).w = ws(i);
endfor
endfunction
% Create array of structures containing quadrature data for integrating over 2D element
%
% e - element index (pair)
% k - quadrature order
% bx, by - 1D bases
%
% returns - array of structures with fields
% x - point
% w - weight
function qs = quad_data2d(e, k, bx, by)
xs = quad_points(bx.points(e(1) + 1), bx.points(e(1) + 2), k);
ys = quad_points(by.points(e(2) + 1), by.points(e(2) + 2), k);
ws = quad_weights(k);
for i = 1:k
for j = 1:k
qs(i, j).x = [xs(i), ys(j)];
qs(i, j).w = ws(i) * ws(j);
endfor
endfor
qs = reshape(qs, 1, []);
endfunction
% Compute quarature data for integrating on selected edges of the 2D element
%
% e - index of the element
% sides - array of 4 boolean values, used to determine which edges to prepare data for.
% Order of the edges:
% 1 - left
% 2 - right
% 3 - top
% 4 - bottom
% k - order of the quadrature
% bx, by - 1D bases
%
% returns - array of structures containing fields:
% jacobian - jacobian of the edge parameterization
% normal - unit vector perpendicular to the edge
% quad_data - points and weights of 1D quadrature on the edge
function es = edge_data(e, sides, k, bx, by)
% Empty structure array
es = struct('jacobian', [], 'normal', [], 'quad_data', []);
if (sides(1))
es(end+1) = edge_data_left(e, k, bx, by);
endif
if (sides(2))
es(end+1) = edge_data_right(e, k, bx, by);
endif
if (sides(3))
es(end+1) = edge_data_top(e, k, bx, by);
endif
if (sides(4))
es(end+1) = edge_data_bottom(e, k, bx, by);
endif
endfunction
% Auxiliary functions - computing quadrature data for each single edge
function edge = edge_data_left(e, k, bx, by)
x1 = bx.points(e(1) + 1);
edge.jacobian = jacobian1d(e(2), by);
edge.normal = [-1 0];
edge.quad_data = quad_data1d(e(2), k, by);
for i = 1:k
edge.quad_data(i).x = [x1, edge.quad_data(i).x];
endfor
endfunction
function edge = edge_data_right(e, k, bx, by)
x2 = bx.points(e(1) + 2);
edge.jacobian = jacobian1d(e(2), by);
edge.normal = [1 0];
edge.quad_data = quad_data1d(e(2), k, by);
for i = 1:k
edge.quad_data(i).x = [x2, edge.quad_data(i).x];
endfor
endfunction
function edge = edge_data_bottom(e, k, bx, by)
y1 = by.points(e(2) + 1);
edge.jacobian = jacobian1d(e(1), bx);
edge.normal = [0 -1];
edge.quad_data = quad_data1d(e(1), k, bx);
for i = 1:k
edge.quad_data(i).x = [edge.quad_data(i).x, y1];
endfor
endfunction
function edge = edge_data_top(e, k, bx, by)
y2 = by.points(e(2) + 2);
edge.jacobian = jacobian1d(e(1), bx);
edge.normal = [0 1];
edge.quad_data = quad_data1d(e(1), k, bx);
for i = 1:k
edge.quad_data(i).x = [edge.quad_data(i).x, y2];
endfor
endfunction
% Modify matrix and right-hand side to enforce uniform (zero) Dirichlet boundary conditions
%
% M - matrix
% F - right-hand side
% dofs - degrees of freedom to be fixed
% bx, by - 1D bases
%
% returns - modified M and F
function [M, F] = dirichlet_bc_uniform(M, F, dofs, bx, by)
for d = dofs
i = linear_index(d, bx, by);
M(i, :) = 0;
M(i, i) = 1;
F(i) = 0;
endfor
endfunction
% Evaluate function on a 2D cartesian product grid
%
% f - function accepting 2D point as a two-element vector
% xs, ys - 1D arrays of coordinates
%
% returns - 2D array of values with (i, j) -> f( xs(j), ys(i) )
% (this order is compatible with plotting functions)
function vals = evaluate_on_grid(f, xs, ys)
[X, Y] = meshgrid(xs, ys);
vals = arrayfun(@(x, y) f([x y]), X, Y);
endfunction
% Subdivide xr and yr into N equal size elements
function [xs, ys] = make_grid(xr, yr, N)
xs = linspace(xr(1), xr(2), N + 1);
ys = linspace(yr(1), yr(2), N + 1);
endfunction
% Plot 2D B-spline with coefficients u on a square given as product of xr and yr
%
% u - matrix of coefficients
% xr, yr - intervals specifying the domain, given as two-element vectors
% N - number of plot 'pixels' in each direction
% bx, by - 1D bases
%
% Domain given by xr and yr should be contained in the domain of the B-spline bases
function surface_plot_spline(u, xr, yr, N, bx, by)
[xs, ys] = make_grid(xr, yr, N);
vals = evaluate_on_grid(@(x) evaluate2d(u, x, bx, by), xs, ys);
surface_plot_values(vals, xs, ys);
endfunction
% Plot arbitrary function on a square given as product of xr and yr
%
% f - function accepting 2D point as a two-element vector
% xr, yr - intervals specifying the domain, given as two-element vectors
% N - number of plot 'pixels' in each direction
function surface_plot_fun(f, xr, yr, N)
[xs, ys] = make_grid(xr, yr, N);
vals = evaluate_on_grid(f, xs, ys);
surface_plot_values(vals, xs, ys);
endfunction
% Plot array of values
%
% vals - 2D array of size [length(ys), length(xs)]
% xs, ys - 1D arrays of coordinates
function surface_plot_values(vals, xs, ys)
surf(xs, ys, vals);
xlabel('x');
ylabel('y');
endfunction
% Function pasting two copies of the knot vector together
function k = repeat_knot(knot, p)
m = max(knot);
k = [knot(1:end-1), knot(p+2:end) + m];
endfunction
% Input data
knot = [0, 0, 0, 1, 2, 2 2]; % knot vector
% Problem formulation
a = @(u, du, v, dv) dot(du, dv);
f = @(x) 3;
g = @(x) (x(1) == -1) * 1 + (x(2) == 1) * (-1);
% Setup
p = degree_from_knot(knot);
k = p + 1;
knot = repeat_knot(knot, p);
points = linspace(-1, 1, max(knot) + 1);
bx = basis1d(p, points, knot);
by = basis1d(p, points, knot);
nx = number_of_dofs(bx);
ny = number_of_dofs(by);
n = nx * ny;
M = sparse(n, n);
F = zeros(n, 1);
idx = @(dof) linear_index(dof, bx, by);
% Assemble the system - matrix and the right-hand side
for e = elements(bx, by)
J = jacobian2d(e, bx, by);
for q = quad_data2d(e, k, bx, by)
basis = basis_evaluator2d(q.x, bx, by);
for i = dofs_on_element2d(e, bx, by)
[v, dv] = basis(i);
for j = dofs_on_element2d(e, bx, by)
[u, du] = basis(j);
M(idx(i), idx(j)) += a(u, du, v, dv) * q.w * J;
endfor
F(idx(i)) += f(q.x) * v * q.w * J;
endfor
endfor
% Boundary integrals
sides = boundary_edges(e, bx, by);
for edge = edge_data(e, sides, k, bx, by)
J = edge.jacobian;
for q = edge.quad_data
basis = basis_evaluator2d(q.x, bx, by);
for i = dofs_on_element2d(e, bx, by)
v = basis(i);
F(idx(i)) += g(q.x) * v * q.w * J;
endfor
endfor
endfor
endfor
% Throw away the unnecessary DoFs - lower left quadrant
cx = floor(nx / 2);
cy = floor(ny / 2);
fixed_dofs = cartesian(0:cx, 0:cy);
[M, F] = dirichlet_bc_uniform(M, F, fixed_dofs, bx, by);
% Solve
u = reshape(M \ F, nx, ny);
% Plot the solution
N = 50;
figure('name', 'Solution', 'Position', [0 0 500 400]);
surface_plot_spline(u, [-1 1], [-1 1], N, bx, by);
Listing 1 (Pobierz): MATLAB code computing the heat transfer over L-shape domain using isogeometric finite element method.
Autorzy kodów w MATLABie: Marcin Łoś i Maciej Woźniak.