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Mesh.m
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Mesh.m
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classdef Mesh < handle
properties
nen % Number of element nodes
ndm % Number of dimensions
numnp % Number of nodal points
numel % Number of elements
ix % Element connectivity matrix
x % Node coordinates matrix
coorSystem % Coordinate system of the mesh
end
methods
% ================
function split(obj, n1, n2, s)
if s < 2
error('Subdivision factor must be >= 2.');
end
Numnp = obj.numnp;
n2el = generateNodesToElements(obj);
n2se = cell(Numnp, 1);
spltEdg = [];
queueElem = {};
i = 1;
while true
if ismember(n2, n2el{n1}{i})
break;
else
i = i + 1;
end
if i > length(n2el{n1})
error('split: invalid (n1,n2)');
end
end
queueElem{end + 1} = n2el{n1}{i};
queueNode = [n1 n2];
while ~isempty(queueElem)
[queueElem, queueNode, spltEdg, n2se, n2el, obj] = ...
ElementSplit(s, queueElem(1), queueNode, spltEdg, n2se, n2el, obj);
end
i = 1;
j = 1;
p = [];
ln2el = size(n2el,1);
while ~isempty(n2el)
currentElement = n2el{i}{1};
found = false;
for elemIndex = 1:size(p,1)
if p(elemIndex,:) == currentElement
found = true;
break;
end
end
if ~found
p(j,:) = currentElement;
j = j + 1;
end
n2el{i}{1} = [];
i = i + 1;
if i > ln2el
for kk = 1:i-1
n2el{kk} = n2el{kk}(~cellfun('isempty', n2el{kk}));
end
n2el = n2el(~cellfun('isempty', n2el));
ln2el = size(n2el,1);
i = 1;
end
end
obj.ix = unique(p, 'rows', 'first');
X = zeros(length(n2el), obj.ndm);
X(1:size(obj.x,1), :) = obj.x;
obj.numel = size(obj.ix,1);
if isempty(obj.coorSystem)
if obj.ndm == 2
obj = setCoordinateSystem1(obj, 'CARTESIAN2D', X);
elseif obj.ndm == 3
obj = setCoordinateSystem1(obj, 'CARTESIAN3D', X);
else
error('split: invalid ndm!');
end
else
for i = 1:Numnp
obj.coorSystem(i) = getFromCartesian(obj, obj.coorSystem(i), X(i, :));
end
end
dfact = 1. / s;
Numnp = Numnp + 1;
for i = 1:3:length(spltEdg)
i1 = min(spltEdg(i), spltEdg(i+1));
i2 = max(spltEdg(i), spltEdg(i+1));
fact1 = 1.;
fact2 = 0.;
for k = 1:s-1
fact1 = fact1 - dfact;
fact2 = fact2 + dfact;
obj.coorSystem(Numnp) = obj.interpolate(obj.coorSystem(i1), obj.coorSystem(i2), fact1, fact2);
X(Numnp,:) = obj.transformToCartesian(obj.coorSystem(Numnp));
Numnp = Numnp + 1;
end
end
obj.x = X;
obj.numnp = size(obj.x,1);
obj.ix = findElementOrder(obj, p, obj.x);
obj.numel = size(obj.ix,1);
fprintf('The element split has been successful.\n');
fprintf(' new numel = %d\n', obj.numel);
fprintf(' new numnp = %d\n\n', obj.numnp);
end
% ================
function n2e = generateNodesToElements(obj)
n2e = cell(obj.numnp, 1);
for i = 1:size(obj.ix,1)
elem = obj.ix(i,:);
for j = 1:obj.nen
nodeIndex = elem(j);
if isempty(n2e{nodeIndex})
n2e{nodeIndex} = {elem};
else
n2e{nodeIndex}{end+1} = elem;
end
end
end
end
% ================
function [queueElem, queueNode, spltEdg, n2se, n2el, obj] = ElementSplit(s, queueElem, queueNode, spltEdg, n2se, n2el, obj)
n1 = queueNode(1);
n2 = queueNode(2);
m = [n1, n2];
ixv = [queueElem{1} queueElem{1}(1)];
for i = 1:5
if isempty(setxor(m, ixv(1:2)))
break;
end
ixv = circshift(ixv, -1);
ixv(end) = ixv(1);
end
n1 = ixv(1);
n2 = ixv(2);
n3 = ixv(3);
n4 = ixv(4);
n12 = zeros(s + 1,1);
n43 = zeros(s + 1,1);
n12(1) = n1;
n43(1) = n4;
n12(s + 1) = n2;
n43(s + 1) = n3;
[k, n2se, n2el, spltEdg, queueElem, queueNode] = splitEdge(obj, n1, n2, s, n2se, n2el, spltEdg, queueElem, queueNode);
for j = 1:s-1
n12(j + 1) = abs(k + j - 1);
end
[k, n2se, n2el, spltEdg, queueElem, queueNode] = splitEdge(obj, n4, n3, s, n2se, n2el, spltEdg, queueElem, queueNode);
for j = 1:s-1
n43(j + 1) = abs(k + j - 1);
end
Ix(1) = n12(1);
Ix(2) = n12(2);
Ix(3) = n43(2);
Ix(4) = n43(1);
bix = queueElem{1};
for i = 1:length(n2el)
for j = 1:length(n2el{i})
if(n2el{i}{j} == bix)
n2el{i}{j} = Ix;
end
end
end
n2el = updateNodesToElements1(obj, n2el,Ix);
for i = 2:s
Ix(1) = n12(i);
Ix(2) = n12(i + 1);
Ix(3) = n43(i + 1);
Ix(4) = n43(i);
n2el = updateNodesToElements1(obj, n2el, Ix);
end
n2el = updateNodesToElements2(obj, n2el, n3);
n2el = updateNodesToElements2(obj, n2el, n4);
queueElem(1) = [];
queueNode(1) = [];
queueNode(1) = [];
end
% ================
function [k, n2se, n2el, spltEdg, queueElem, queueNode] = splitEdge(~, n1, n2, s, n2se, n2el, spltEdg, queueElem, queueNode)
nmn = min(n1, n2);
nmx = max(n1, n2);
for i = 1:length(n2se{nmn})
if(spltEdg(n2se{nmn}(i) + 2) == nmx)
k = spltEdg(n2se{nmn}(i) + 3);
if(n1 >= n2); k = 2 - k - s; end
return;
end
end
k = length(n2se) + 1;
n2se{k + s - 2, 1} = [];
n2se{nmn} = [n2se{nmn} length(spltEdg)];
spltEdg = [spltEdg nmn nmx k];
for i = 1:length(n2el{n1})
elemIndex = n2el{n1}{i};
contains = any(cellfun(@(x) isequal(x, elemIndex), queueElem));
if ~contains
if ismember(n2, elemIndex)
queueElem{end + 1} = elemIndex;
queueNode = [queueNode, n1, n2];
end
end
end
if n1 >= n2
k = 2 - k - s;
end
end
% ================
function n2e = updateNodesToElements1(~, n2e, ix)
for i = 1:length(ix)
k = ix(i);
if (k > length(n2e))
for j = (length(n2e) + 1):k
n2e{j,1} = {};
end
end
n2e{k} = [n2e{k} ix];
end
end
% ================
function n2e = updateNodesToElements2(~, n2e, n)
i = 1;
while i <= length(n2e{n})
if ~any(n2e{n}{i} == n)
n2e{n}(i) = [];
else
i = i + 1;
end
end
end
% ================
function obj = setCoordinateSystem1(obj, t, x)
Numnp = obj.numnp;
obj.coorSystem = repmat(struct('type', '', 'xc', [], 'coor', []), Numnp, 1);
for i = 1:Numnp
obj.coorSystem(i).type = t;
obj.coorSystem(i) = getFromCartesian(obj, obj.coorSystem(i), x(i, :));
end
end
% ================
function elem = getFromCartesian(~, elem, x)
switch elem.type
case 'CARTESIAN3D'
elem.coor = x(1:3);
case 'CARTESIAN2D'
elem.coor = x(1:2);
case 'POLAR'
dx = x(1) - elem.xc(1);
dy = x(2) - elem.xc(2);
elem.coor = [atan2(dy, dx), sqrt(dx^2 + dy^2)];
case 'SPHERICAL'
dx = x(1) - elem.xc(1);
dy = x(2) - elem.xc(2);
dz = x(3) - elem.xc(3);
elem.coor = [sqrt(dx^2 + dy^2 + dz^2), atan2(dy, dx), atan2(sqrt(dx^2 + dy^2), dz)];
end
end
% ================
function cs = interpolate(obj, c1, c2, f1, f2)
if strcmp(c1.type, c2.type)
coor1 = [c1.coor(1), c1.coor(2)];
coor2 = [c2.coor(1), c2.coor(2)];
type = c1.type;
if strcmp(type, 'SPHERICAL')
error('SPHERICAL interpolation is not implemented.');
end
if strcmp(type, 'POLAR') || strcmp(type, 'CYLINDRICAL')
if coor2(1) + pi + 1e-4 < coor1(1)
coor2(1) = coor2(1) + 2*pi;
elseif coor1(1) + pi + 1e-4 < coor2(1)
coor1(1) = coor1(1) + 2*pi;
end
end
coor = f1 * coor1 + f2 * coor2;
xc = c1.xc;
else
x1 = obj.transformToCartesian(c1);
x2 = obj.transformToCartesian(c2);
coor = zeros(1, 2);
for i = 1:2
coor(i) = f1 * x1(i) + f2 * x2(i);
end
if strcmp(c1.type, 'POLAR') || strcmp(c1.type, 'CARTESIAN2D') || strcmp(c2.type, 'POLAR') || strcmp(c2.type, 'CARTESIAN2D')
type = 'CARTESIAN2D';
else
type = 'CARTESIAN3D';
end
end
cs.type = type;
cs.coor = coor;
cs.xc = [ 0, 0];
end
% ================
function x = transformToCartesian(~, csElement)
switch csElement.type
case 'CARTESIAN3D'
x(1) = csElement.coor(1);
x(2) = csElement.coor(2);
x(3) = csElement.coor(3);
case 'CARTESIAN2D'
x(1) = csElement.coor(1);
x(2) = csElement.coor(2);
case 'CYLINDRICAL'
x(1) = csElement.xc(1) + cos(csElement.coor(1)) * csElement.coor(2);
x(2) = csElement.xc(2) + sin(csElement.coor(1)) * csElement.coor(2);
x(3) = csElement.coor(3);
case 'POLAR'
x(1) = csElement.xc(1) + cos(csElement.coor(1)) * csElement.coor(2);
x(2) = csElement.xc(2) + sin(csElement.coor(1)) * csElement.coor(2);
case 'SPHERICAL'
x(1) = csElement.xc(1) + cos(csElement.coor(2)) * sin(csElement.coor(3)) * csElement.coor(1);
x(2) = csElement.xc(2) + sin(csElement.coor(2)) * sin(csElement.coor(3)) * csElement.coor(1);
x(3) = csElement.xc(3) + cos(csElement.coor(3)) * csElement.coor(1);
end
end
% ================
function orderedElements = findElementOrder(~, elements, nodeCoords)
numElements = size(elements, 1);
centroids = zeros(numElements, 2);
for i = 1:numElements
elementNodes = elements(i, :);
xMean = mean(nodeCoords(elementNodes, 1));
yMean = mean(nodeCoords(elementNodes, 2));
centroids(i, :) = [xMean, yMean];
end
[~, sortOrder] = sortrows(centroids, [2, 1]);
elementsSortedByRow = elements(sortOrder, :);
centroidsSortedByRow = centroids(sortOrder, :);
uniqueY = unique(centroidsSortedByRow(:, 2));
orderedElements = [];
for i = 1:length(uniqueY)
rowIndices = find(centroidsSortedByRow(:, 2) == uniqueY(i));
elementsInRow = elementsSortedByRow(rowIndices, :);
[~, rowSortOrder] = sort(centroidsSortedByRow(rowIndices, 1));
if mod(i, 2) == 0
rowSortOrder = rowSortOrder(end:-1:1);
end
orderedElements = [orderedElements; elementsInRow(rowSortOrder, :)];
end
end
% ================
function obj = smoothInternal(obj, iter)
Numnp = obj.numnp;
X = obj.x;
n2n = generateNodesToNodes(obj);
if (obj.ndm == 2)
bndNodeToNode = ComputeBoundaryData2D(obj);
elseif(obj.ndm == 3)
error('smoothInternal: 3D not implemented');
else
error('smoothInternal: 2D and 3D only!');
end
for it = 1:iter
for i = 1:Numnp
if isempty(bndNodeToNode{i}(:))
xi = X(n2n{i}(1), :);
for j = 2:length(n2n{i})
xj = X(n2n{i}(j), :);
xi = xi + xj;
end
fact = 1. / length(n2n{i});
xi = xi * fact;
X(i, :) = xi;
end
end
end
obj.x = X;
end
% ================
function n2n = generateNodesToNodes(obj)
Ix = obj.ix;
numNodes = max(Ix(:));
n2n = cell(numNodes, 1);
for elem = 1:size(Ix, 1)
for n = 1:size(Ix, 2)
currentNode = Ix(elem, n);
n2n{currentNode} = unique([n2n{currentNode}, Ix(elem, Ix(elem, :) ~= currentNode)]);
end
end
end
% ================
function bndNodeToNode = ComputeBoundaryData2D(obj)
Numnp = obj.numnp;
Numel = obj.numel;
tmp = cell(1, Numnp);
for elemIndex = 1:Numel
edge = getMeshEdges(obj, obj.ix(elemIndex, :), obj.nen);
for i = 1:2:length(edge)
n1 = edge(i);
n2 = edge(i + 1);
if ~isempty(tmp{n2}) && any(tmp{n2} == n1)
tmp{n2}(tmp{n2} == n1) = [];
else
tmp{n1}(end+1) = n2;
end
end
end
bndNodeToNode = cell(size(tmp));
for i = 1:Numnp
for j = 1:length(tmp{i})
bndNodeToNode{i} = [ bndNodeToNode{i}, tmp{i}(j)];
bndNodeToNode{tmp{i}(j)} = [ bndNodeToNode{tmp{i}(j)}, i];
end
end
end
% ================
function edge = getMeshEdges(~, ix, nen)
switch nen
case 3
edge = [ix(1), ix(2), ix(2), ix(3), ix(3), ix(1)];
case 4
edge = [ix(1), ix(2), ix(2), ix(3), ix(3), ix(4), ix(4), ix(1)];
otherwise
error('getMeshEdges: Element Type not implemented');
end
end
% ================
function obj = setCoordinateSystem(obj, node, t, xx)
Numnp = obj.numnp;
if (isempty(obj.coorSystem) || length(obj.coorSystem) < Numnp)
obj.coorSystem = repmat(struct('type', '', 'xc', xx, 'coor', []), Numnp, 1);
if obj.ndm == 2
for i = 1:Numnp
obj.coorSystem(i).type = 'CARTESIAN2D';
end
elseif obj.ndm == 3
for i = 1:Numnp
obj.coorSystem(i).type = 'CARTESIAN3D';
end
end
for i = 1:Numnp
obj.coorSystem(i) = getFromCartesian(obj, obj.coorSystem(i),obj.x(i,:));
end
end
for i = 1:length(node)
if ~isempty(xx)
for j = 1:obj.ndm
obj.coorSystem(node(i)).xc(j) = xx(j);
end
end
obj.coorSystem(node(i)) = getFromCartesian(obj, obj.coorSystem(i),obj.x(node(i),:));
obj.coorSystem(node(i)).type = t;
end
end
% ================
function plotMesh(obj, varargin)
showNodes = false;
showElements = false;
for i = 1:length(varargin)
if strcmp(varargin{i}, 'nodes')
showNodes = true;
elseif strcmp(varargin{i}, 'elements')
showElements = true;
end
end
patch('Vertices', obj.x, 'Faces', obj.ix, 'FaceColor', 'white', 'EdgeColor', 'black');
hold on;
if showNodes
plot(obj.x(:,1), obj.x(:,2), 'ro');
for i = 1:size(obj.x, 1)
text(obj.x(i,1), obj.x(i,2), sprintf('%d', i), 'VerticalAlignment', 'bottom', 'HorizontalAlignment', 'right', 'Color', 'red', 'Clipping', 'on');
end
end
if showElements
for i = 1:size(obj.ix, 1)
centroid = mean(obj.x(obj.ix(i,:), :), 1);
text(centroid(1), centroid(2), sprintf('%d', i), ...
'VerticalAlignment', 'cap', 'HorizontalAlignment', 'center', 'Color', 'blue', 'Clipping', 'on');
end
end
hold off;
axis equal;
xRange = range(obj.x(:,1));
yRange = range(obj.x(:,2));
maxRange = max(xRange, yRange);
margin = maxRange * 0.05;
xlim([min(obj.x(:,1)) - margin, max(obj.x(:,1)) + margin]);
ylim([min(obj.x(:,2)) - margin, max(obj.x(:,2)) + margin]);
end
% ================
end
end