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Example6.java

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package symjava.examples;
import static symjava.math.SymMath.dot;
import static symjava.math.SymMath.grad;
import static symjava.symbolic.Symbol.*;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import Jama.Matrix;
import symjava.domains.Domain;
import symjava.examples.fem.Element;
import symjava.examples.fem.Mesh2D;
import symjava.examples.fem.Node;
import symjava.examples.fem.RefTriangle;
import symjava.examples.fem.WeakForm;
import symjava.math.Transformation;
import symjava.matrix.SymMatrix;
import symjava.numeric.NumInt;
import symjava.relational.Eq;
import symjava.symbolic.Expr;
import symjava.symbolic.Func;
import symjava.symbolic.Integrate;
import symjava.symbolic.SymConst;
import symjava.symbolic.Symbol;
/**
* Finite Element solver
*
*/
public class Example6 {
public static void main(String[] args) {
// Func u = new Func("u", x, y);
// Func v = new Func("v", x, y);
//
//// //Our PDE equation
//// Eq pde = new Eq(0.5*dot(grad(u), grad(v)) + 0.1*u*v, (x*x+y*y)*v);
//// //Read the mesh
//// Mesh2D mesh = new Mesh2D("mesh1", x, y);
//// mesh.readTriangleMesh("double_hex3.1.node", "double_hex3.1.ele");
//// solve(pde, mesh, null, "double_hex3.1.dat");
//
// //Another PDE equation with Dirichlet condition
// WeakForm wf = new WeakForm(dot(grad(u), grad(v)), (-2*(x*x+y*y)+36)*v, u, v);
// //Eq pde2 = new Eq(u*v, (-2*(x*x+y*y)+36)*v);
// Mesh2D mesh2 = new Mesh2D("mesh2");
// //mesh2.readGridGenMesh("patch_triangle.grd");
// mesh2.readGridGenMesh("triangle.grd");
// //Mark boundary nodes
// double eps = 0.01;
// for(Node n : mesh2.nodes) {
// //if(1-Math.abs(n.coords[0])<eps || 1-Math.abs(n.coords[1])<eps || Math.abs(n.coords[0])<eps || Math.abs(n.coords[1])<eps )
// if(Math.abs(3-Math.abs(n.coords[0]))<eps || Math.abs(3-Math.abs(n.coords[1]))<eps)
// n.setType(1);
// }
// Map<Integer, Double> diri = new HashMap<Integer, Double>();
// diri.put(1, 0.0);
// //solve(pde2, mesh2, diri, "patch_triangle.dat");
// solve(wf, mesh2, diri, "triangle.dat");
// solve2(mesh2, diri, "triangle_hardcode.dat");
peper_example();
}
public static void peper_example() {
Func u = new Func("u", x, y);
Func v = new Func("v", x, y);
WeakForm wf = new WeakForm(dot(grad(u), grad(v)), (-2*(x*x+y*y)+36)*v, u, v);
Mesh2D mesh = new Mesh2D("Square");
mesh.readGridGenMesh("triangle.grd");
//Mark boundary nodes
double eps = 0.01;
for(Node n : mesh.nodes) {
if(Math.abs(3-Math.abs(n.coords[0]))<eps || Math.abs(3-Math.abs(n.coords[1]))<eps)
n.setType(1);
}
Map<Integer, Double> diri = new HashMap<Integer, Double>();
diri.put(1, 0.0);
solve(wf, mesh, diri, "triangle.dat");
}
public static void solve(WeakForm wf, Mesh2D mesh, Map<Integer, Double> dirichlet, String output) {
System.out.println(String.format("PDE Weak Form: %s = %s", wf.lhs(), wf.rhs()));
//Create coordinate transformation
Symbol x1 = new Symbol("x1");
Symbol x2 = new Symbol("x2");
Symbol x3 = new Symbol("x3");
Symbol y1 = new Symbol("y1");
Symbol y2 = new Symbol("y2");
Symbol y3 = new Symbol("y3");
Transformation trans = new Transformation(
new Eq(x, x1*r+x2*s+x3*(1-r-s), new Expr[]{r, s}),
new Eq(y, y1*r+y2*s+y3*(1-r-s), new Expr[]{r, s})
);
// jac = (xr xs)
// (yr ys)
SymMatrix jacMat = trans.getJacobianMatrix();
System.out.println(jacMat);
System.out.println();
//Shape functions
Func N1 = new Func("R", x, y);
Func N2 = new Func("S", x, y);
Func N3 = new Func("T", 1 - N1 - N2);
Func[] shapeFuns = {N1, N2, N3};
Expr jac = trans.getJacobian();
Expr rx = jacMat[1][1]/jac; //rx = ys/jac
Expr ry = -jacMat[0][1]/jac; //ry = -xs/jac //bugfix missed a minus sign!
Expr sx = -jacMat[1][0]/jac; //sx = -yr/jac
Expr sy = jacMat[0][0]/jac; //sy = xr/jac
System.out.println(jac);
System.out.println(rx);
System.out.println(ry);
System.out.println(sx);
System.out.println(sy);
RefTriangle tri = new RefTriangle("Tri", r, s);
Integrate lhsInt[][] = new Integrate[shapeFuns.length][shapeFuns.length];
Integrate rhsInt[] = new Integrate[shapeFuns.length];
for(int i=0; i<shapeFuns.length; i++) {
Func V = shapeFuns[i]; //test
for(int j=0; j<shapeFuns.length; j++) {
Func U = shapeFuns[j]; //trial
//Weak form for the left hand side of the PDE
Expr lhs = wf.lhs().subs(wf.trial, U).subs(wf.test, V);
System.out.println(lhs);
System.out.println();
//Replace the derivatives with it's concrete expression
lhs = lhs.subs(N1.diff(x), rx)
.subs(N1.diff(y), ry)
.subs(N2.diff(x), sx)
.subs(N2.diff(y), sy)
.subs(N1, r).subs(N2, s)
.subs(x, trans.eqs[0].rhs())
.subs(y, trans.eqs[1].rhs());
System.out.println(lhs);
System.out.println();
//Define the integration on the reference domain
//lhsInt[i][j] = new Int(new Func("",lhs*jac,new Expr[]{r,s}), tri);
lhsInt[i][j] = new Integrate(new Func(
String.format("lhs%d%d",i,j), lhs*jac,new Expr[]{r,s}), tri);
System.out.println(lhsInt[i][j]);
System.out.println();
}
//Weak form for the right hand side of the PDE
Expr rhs = wf.rhs().subs(wf.test, V)
.subs(N1, r).subs(N2, s)
.subs(x, trans.eqs[0].rhs())
.subs(y, trans.eqs[1].rhs());
//System.out.println(rhs);
System.out.println();
rhsInt[i] = new Integrate(new Func(
String.format("rhs%d",i),rhs*jac,new Expr[]{r,s}), tri);
}
// Expr ttt = lhsInt[0][1].integrand
// .subs(x1, -1.996794872)
// .subs(x2, -1.786508761)
// .subs(x3, -1.381601534)
// .subs(y1, 1.996794872)
// .subs(y2, 1.395640906)
// .subs(y3, 1.747001174).simplify();
// System.out.println(ttt);
// Func fttt = new Func("fttt",ttt);
// BytecodeFunc bfttt = fttt.toBytecodeFunc();
// System.out.println(bfttt.apply());
//Generate bytecode for the integration
//You can save the class to some place for later use
NumInt lhsNInt[][] = new NumInt[shapeFuns.length][shapeFuns.length];
NumInt rhsNInt[] = new NumInt[shapeFuns.length];
for(int i=0; i<shapeFuns.length; i++) {
for(int j=0; j<shapeFuns.length; j++) {
lhsNInt[i][j] = new NumInt(lhsInt[i][j]);
}
rhsNInt[i] = new NumInt(rhsInt[i]);
}
//Assemble the system
System.out.println("Start assemble the system...");
long begin = System.currentTimeMillis();
List<Domain> eles = mesh.getSubDomains();
double[][] matA = new double[mesh.nodes.size()][mesh.nodes.size()];
double[] vecb = new double[mesh.nodes.size()];
for(Domain d : eles) {
Element e = (Element)d;
double[] nodeCoords = e.getNodeCoords();
for(int i=0; i<shapeFuns.length; i++) {
int idxI = e.nodes.get(i).getIndex()-1;
for(int j=0; j<shapeFuns.length; j++) {
int idxJ = e.nodes.get(j).getIndex()-1;
double t = lhsNInt[i][j].eval(nodeCoords);
//System.out.println(idxI+" "+idxJ+" "+t);
matA[idxI][idxJ] += t;
}
vecb[idxI] += rhsNInt[i].eval(nodeCoords);
}
}
System.out.println("Assemble done! Time: "+(System.currentTimeMillis()-begin)+"ms");
System.out.println("Solving...");
begin = System.currentTimeMillis();
Matrix A = new Matrix(matA);
Matrix b = new Matrix(vecb, vecb.length);
if(dirichlet != null) {
for(Node n : mesh.nodes) {
Double diri = dirichlet.get(n.getType());
if(diri != null) {
setDirichlet(A, b, n.getIndex()-1, diri);
}
}
}
// for(int i=100; i<=110; i++) {
// for(int j=100; j<=110; j++)
// System.out.print(A.get(i-1, j-1)+" ");
// System.out.println();
// }
Matrix x = A.solve(b);
// for(int i=0; i<x.getRowDimension(); i++)
// System.out.println(x.get(i, 0));
mesh.writeTechplot(output, x.getArray());
System.out.println("Solved! Time: "+(System.currentTimeMillis()-begin)+"ms");
System.out.println("See the output file(Tecplot format) "+output+" for the solution.");
}
public static void setDirichlet(Matrix A, Matrix b, int nodeIndex, double value) {
int row = nodeIndex;
int col = nodeIndex;
A.set(row, col, 1.0);
b.set(row, 0, value);
for(int r=0; r<A.getRowDimension(); r++) {
if(r != row) {
A.set(r, col, 0.0);
b.set(r, 0 , b.get(r, 0)-A.get(r, col)*value);
}
}
for(int c=0; c<A.getColumnDimension(); c++) {
if(c != col) {
A.set(row, c, 0.0);
}
}
}
/**
* Hard code the solver for
*
* dot(grad(u), grad(v)) == (-2*(x*x+y*y)+36)*v)
*
* @param mesh
* @param dirichlet
* @param output
*/
public static void solve2(Mesh2D mesh, Map<Integer, Double> dirichlet, String output) {
//Assemble the system
System.out.println("Start assemble the system...");
long begin = System.currentTimeMillis();
List<Domain> eles = mesh.getSubDomains();
double[][] matA = new double[mesh.nodes.size()][mesh.nodes.size()];
double[] vecb = new double[mesh.nodes.size()];
for(Domain d : eles) {
Element e = (Element)d;
double[] coords = e.getNodeCoords();
double x1 = coords[0];
double x2 = coords[1];
double x3 = coords[2];
double y1 = coords[3];
double y2 = coords[4];
double y3 = coords[5];
//x = x1*r+x2*s+x3*(1-r-s)
//y = y1*r+y2*s+y3*(1-r-s)
double xr = x1-x3;
double xs = x2-x3;
double yr = y1-y3;
double ys = y2-y3;
// jac = (xr xs)
// (yr ys)
double jac = xr*ys-xs*yr;
double rx = ys/jac;
double ry = -xs/jac;
double sx = -yr/jac;
double sy = xr/jac;
double tx = -rx - sx;
double ty = -ry - sy;
double[][] lhs = {
{rx*rx + ry*ry, rx*sx + ry*sy, rx*tx+ry*ty},
{sx*rx + sy*ry, sx*sx + sy*sy, sx*tx+sy*ty},
{tx*rx + ty*ry, tx*sx + ty*sy, tx*tx+ty*ty},
};
for(int i=0; i<3; i++) {
int idxI = e.nodes.get(i).getIndex()-1;
for(int j=0; j<3; j++) {
int idxJ = e.nodes.get(j).getIndex()-1;
double t = lhs[i][j]*jac*0.5;
matA[idxI][idxJ] += t;
}
}
double[][] intPnW = {
{0.5, 0.5, 1.0/6.0},
{0.0, 0.5, 1.0/6.0},
{0.5, 0.0, 1.0/6.0}
};
for(int k=0; k<intPnW.length; k++) {
double r=intPnW[k][0];
double s=intPnW[k][1];
double t=1-r-s;
double w=intPnW[k][2];
double x = x1*r+x2*s+x3*t;
double y = y1*r+y2*s+y3*t;
//f=-2*(x*x+y*y)+36
double f = -2.0*(x*x+y*y)+36.0;
vecb[e.nodes.get(0).getIndex() - 1] += f*r*jac*w;
vecb[e.nodes.get(1).getIndex() - 1] += f*s*jac*w;
vecb[e.nodes.get(2).getIndex() - 1] += f*t*jac*w;
}
}
System.out.println("Assemble done! Time: "+(System.currentTimeMillis()-begin)+"ms");
System.out.println("Solving...");
begin = System.currentTimeMillis();
Matrix A = new Matrix(matA);
Matrix b = new Matrix(vecb, vecb.length);
if(dirichlet != null) {
for(Node n : mesh.nodes) {
Double diri = dirichlet.get(n.getType());
if(diri != null) {
setDirichlet(A, b, n.getIndex()-1, diri);
}
}
}
// for(int i=100; i<=110; i++) {
// for(int j=100; j<=110; j++)
// System.out.print(A.get(i-1, j-1)+" ");
// System.out.println();
// }
// for(int i=100; i<=110; i++) {
// System.out.println(b.get(i-1, 0));
// }
/*
3.678403159123969 -0.9973611181090696 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
-0.9973611181090696 3.877353541972138 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 3.525365130978284 -0.4486227282431723 -0.393018578918151 0.0 0.0 0.0 0.0 -0.24553903148664002 0.0
0.0 0.0 -0.4486227282431723 3.5372167175010087 -0.8194573851545361 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 -0.393018578918151 -0.8194573851545361 3.535701918507291 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 3.4945816438189032 -0.6132348151677102 -0.22191378338605314 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 -0.6132348151677102 3.707317244324667 -0.8671907304126691 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 -0.22191378338605314 -0.8671907304126691 3.4758034494659844 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.623619648286562 0.0 0.0
0.0 0.0 -0.24553903148664002 0.0 0.0 0.0 0.0 0.0 0.0 3.501878694448894 -0.4577721650038804
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.4577721650038804 3.6326693760208175
5.920480649794939
5.830543653385974
11.166016717821526
9.51298710892804
7.987159886596924
9.00295976201226
6.471980478636926
9.623214010483725
6.664903619294702
10.961199858227298
7.935696861306976
*/
Matrix x = A.solve(b);
mesh.writeTechplot(output, x.getArray());
System.out.println("Solved! Time: "+(System.currentTimeMillis()-begin)+"ms");
System.out.println("See the output file(Tecplot format) "+output+" for the solution.");
}
}
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