How do I use low level FEM?
$begingroup$
How do I simulate the following partial differential equation using Low level FEM in Mathematica?
D[u[x,y], x] - D[u[x,y], y] = x Sin[x y] - y Sin[x y]
The answer is Cos[x y]
added)
The region is [-1,1] for x and [-1,1] for y.
The boundary condition is u[-1,y]=Cos[-y]
Here is my tried code:
delta[x_, a_] := Block[{}, If[x == a, 1, 0]];
mesh = Table[PDF[NormalDistribution[a, z], x]*
PDF[NormalDistribution[b, z], y], {a, -1, 1, 0.1},
{b, -1, 1, 0.1}];
mesh = Flatten[mesh];
cofs = Table[ToExpression[StringJoin[ToString[c],
ToString[a], ToString[b]]], {a, 0, 20}, {b, 0, 20}];
cofs = Flatten[cofs];
weights = Table[delta[x, a]*delta[y, b], {a, -1, 1, 0.05},
{b, -1, 1, 0.05}];
weights = Flatten[weights];
expr = cofs . mesh;
residual = D[expr, x] - D[expr, y] - (x*Sin[x*y] -
y*Sin[x*y]);
integrated = residual*Plus @@ weights;
toMinimize = Plus @@ Flatten[Table[integrated,
{x, -1, 1, 0.1}, {y, -1, 1, 0.1}]];
toMinimize = toMinimize^2;
cofs = Append[cofs, z];
conds = Abs[(expr /. {x -> 0, y -> 0}) - 1];
opt = NMinimize[{toMinimize + conds}, cofs,
Method -> {Automatic, PerformanceGoal -> "Speed"}];
u = expr /. Last[opt];
ListPlot3D[Table[u*Plus @@ weights /. {x -> a, y -> b},
{a, -1, 1, 0.1}, {b, -1, 1, 0.1}]]
finite-element-method
$endgroup$
|
show 5 more comments
$begingroup$
How do I simulate the following partial differential equation using Low level FEM in Mathematica?
D[u[x,y], x] - D[u[x,y], y] = x Sin[x y] - y Sin[x y]
The answer is Cos[x y]
added)
The region is [-1,1] for x and [-1,1] for y.
The boundary condition is u[-1,y]=Cos[-y]
Here is my tried code:
delta[x_, a_] := Block[{}, If[x == a, 1, 0]];
mesh = Table[PDF[NormalDistribution[a, z], x]*
PDF[NormalDistribution[b, z], y], {a, -1, 1, 0.1},
{b, -1, 1, 0.1}];
mesh = Flatten[mesh];
cofs = Table[ToExpression[StringJoin[ToString[c],
ToString[a], ToString[b]]], {a, 0, 20}, {b, 0, 20}];
cofs = Flatten[cofs];
weights = Table[delta[x, a]*delta[y, b], {a, -1, 1, 0.05},
{b, -1, 1, 0.05}];
weights = Flatten[weights];
expr = cofs . mesh;
residual = D[expr, x] - D[expr, y] - (x*Sin[x*y] -
y*Sin[x*y]);
integrated = residual*Plus @@ weights;
toMinimize = Plus @@ Flatten[Table[integrated,
{x, -1, 1, 0.1}, {y, -1, 1, 0.1}]];
toMinimize = toMinimize^2;
cofs = Append[cofs, z];
conds = Abs[(expr /. {x -> 0, y -> 0}) - 1];
opt = NMinimize[{toMinimize + conds}, cofs,
Method -> {Automatic, PerformanceGoal -> "Speed"}];
u = expr /. Last[opt];
ListPlot3D[Table[u*Plus @@ weights /. {x -> a, y -> b},
{a, -1, 1, 0.1}, {b, -1, 1, 0.1}]]
finite-element-method
$endgroup$
$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
1
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago
|
show 5 more comments
$begingroup$
How do I simulate the following partial differential equation using Low level FEM in Mathematica?
D[u[x,y], x] - D[u[x,y], y] = x Sin[x y] - y Sin[x y]
The answer is Cos[x y]
added)
The region is [-1,1] for x and [-1,1] for y.
The boundary condition is u[-1,y]=Cos[-y]
Here is my tried code:
delta[x_, a_] := Block[{}, If[x == a, 1, 0]];
mesh = Table[PDF[NormalDistribution[a, z], x]*
PDF[NormalDistribution[b, z], y], {a, -1, 1, 0.1},
{b, -1, 1, 0.1}];
mesh = Flatten[mesh];
cofs = Table[ToExpression[StringJoin[ToString[c],
ToString[a], ToString[b]]], {a, 0, 20}, {b, 0, 20}];
cofs = Flatten[cofs];
weights = Table[delta[x, a]*delta[y, b], {a, -1, 1, 0.05},
{b, -1, 1, 0.05}];
weights = Flatten[weights];
expr = cofs . mesh;
residual = D[expr, x] - D[expr, y] - (x*Sin[x*y] -
y*Sin[x*y]);
integrated = residual*Plus @@ weights;
toMinimize = Plus @@ Flatten[Table[integrated,
{x, -1, 1, 0.1}, {y, -1, 1, 0.1}]];
toMinimize = toMinimize^2;
cofs = Append[cofs, z];
conds = Abs[(expr /. {x -> 0, y -> 0}) - 1];
opt = NMinimize[{toMinimize + conds}, cofs,
Method -> {Automatic, PerformanceGoal -> "Speed"}];
u = expr /. Last[opt];
ListPlot3D[Table[u*Plus @@ weights /. {x -> a, y -> b},
{a, -1, 1, 0.1}, {b, -1, 1, 0.1}]]
finite-element-method
$endgroup$
How do I simulate the following partial differential equation using Low level FEM in Mathematica?
D[u[x,y], x] - D[u[x,y], y] = x Sin[x y] - y Sin[x y]
The answer is Cos[x y]
added)
The region is [-1,1] for x and [-1,1] for y.
The boundary condition is u[-1,y]=Cos[-y]
Here is my tried code:
delta[x_, a_] := Block[{}, If[x == a, 1, 0]];
mesh = Table[PDF[NormalDistribution[a, z], x]*
PDF[NormalDistribution[b, z], y], {a, -1, 1, 0.1},
{b, -1, 1, 0.1}];
mesh = Flatten[mesh];
cofs = Table[ToExpression[StringJoin[ToString[c],
ToString[a], ToString[b]]], {a, 0, 20}, {b, 0, 20}];
cofs = Flatten[cofs];
weights = Table[delta[x, a]*delta[y, b], {a, -1, 1, 0.05},
{b, -1, 1, 0.05}];
weights = Flatten[weights];
expr = cofs . mesh;
residual = D[expr, x] - D[expr, y] - (x*Sin[x*y] -
y*Sin[x*y]);
integrated = residual*Plus @@ weights;
toMinimize = Plus @@ Flatten[Table[integrated,
{x, -1, 1, 0.1}, {y, -1, 1, 0.1}]];
toMinimize = toMinimize^2;
cofs = Append[cofs, z];
conds = Abs[(expr /. {x -> 0, y -> 0}) - 1];
opt = NMinimize[{toMinimize + conds}, cofs,
Method -> {Automatic, PerformanceGoal -> "Speed"}];
u = expr /. Last[opt];
ListPlot3D[Table[u*Plus @@ weights /. {x -> a, y -> b},
{a, -1, 1, 0.1}, {b, -1, 1, 0.1}]]
finite-element-method
finite-element-method
edited 10 mins ago
Peter Mortensen
32627
32627
asked 8 hours ago
XminerXminer
766
766
$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
1
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago
|
show 5 more comments
$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
1
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago
$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
1
1
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
I am not a 100% sure I understand your question, here is what I think could help you.
The finite element method is, strictly speaking, not a method to solve PDEs. What it does it takes a continuous PDE and converts it to an approximate equivalent matrix and vector. The matrix and vector are discrete representations of the PDE. When you solve this set of equations you will get an approximate solution to the PDE.
We set up an equation:
Needs["NDSolve`FEM`"]
{state} =
NDSolve`ProcessEquations[{Laplacian[u[x, y], {x, y}] == 1,
DirichletCondition[u[x, y] == 0, True]}, u, {x, 0, 1}, {y, 0, 1},
Method -> {"FiniteElement"}];
Now we extract some data from the NDSolve state data object.
femdata = state["FiniteElementData"]
femdata["Properties"]
methodData = femdata["FEMMethodData"];
bcData = femdata["BoundaryConditionData"];
pdeData = femdata["PDECoefficientData"];
variableData = state["VariableData"];
solutionData = state["SolutionData"][[1]];
If you do not want to / can not use NDSolve`ProcessEquations you may need to look at InitializePDECoefficients
and such functions.
If you look at pdeData, that now contains the coefficients of the equations given in NDSolve:
pdeData["All"]
{{{{1}}, {{{{0}, {0}}}}}, {{{{{-1,
0}, {0, -1}}}}, {{{{0}, {0}}}}, {{{{0,
0}}}}, {{0}}}, {{{0}}}, {{{0}}}}
Now, the finite element method is applied. This converts the continuous PDE into a system of discrete matrices:
discretePDE = DiscretizePDE[pdeData, methodData, solutionData]
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"]
You can look at the matrices:
MatrixPlot[stiffness]
The same conversion is done for the boundary conditions:
discreteBCs =
DiscretizeBoundaryConditions[bcData, methodData, solutionData];
All the finite element method does is done now. The rest is linear algebra.
We now put the discrete boundary conditions into the discrete matrices:
DeployBoundaryConditions[{load, stiffness}, discreteBCs]
Solve:
solution = LinearSolve[stiffness, load];
Generate an Interpolating function:
mesh = methodData["ElementMesh"];
ifun = ElementMeshInterpolation[{mesh}, solution]
And visualize:
Plot3D[ifun[x, y], {x, y} [Element] mesh]
Hope this helps a bit.
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I am not a 100% sure I understand your question, here is what I think could help you.
The finite element method is, strictly speaking, not a method to solve PDEs. What it does it takes a continuous PDE and converts it to an approximate equivalent matrix and vector. The matrix and vector are discrete representations of the PDE. When you solve this set of equations you will get an approximate solution to the PDE.
We set up an equation:
Needs["NDSolve`FEM`"]
{state} =
NDSolve`ProcessEquations[{Laplacian[u[x, y], {x, y}] == 1,
DirichletCondition[u[x, y] == 0, True]}, u, {x, 0, 1}, {y, 0, 1},
Method -> {"FiniteElement"}];
Now we extract some data from the NDSolve state data object.
femdata = state["FiniteElementData"]
femdata["Properties"]
methodData = femdata["FEMMethodData"];
bcData = femdata["BoundaryConditionData"];
pdeData = femdata["PDECoefficientData"];
variableData = state["VariableData"];
solutionData = state["SolutionData"][[1]];
If you do not want to / can not use NDSolve`ProcessEquations you may need to look at InitializePDECoefficients
and such functions.
If you look at pdeData, that now contains the coefficients of the equations given in NDSolve:
pdeData["All"]
{{{{1}}, {{{{0}, {0}}}}}, {{{{{-1,
0}, {0, -1}}}}, {{{{0}, {0}}}}, {{{{0,
0}}}}, {{0}}}, {{{0}}}, {{{0}}}}
Now, the finite element method is applied. This converts the continuous PDE into a system of discrete matrices:
discretePDE = DiscretizePDE[pdeData, methodData, solutionData]
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"]
You can look at the matrices:
MatrixPlot[stiffness]
The same conversion is done for the boundary conditions:
discreteBCs =
DiscretizeBoundaryConditions[bcData, methodData, solutionData];
All the finite element method does is done now. The rest is linear algebra.
We now put the discrete boundary conditions into the discrete matrices:
DeployBoundaryConditions[{load, stiffness}, discreteBCs]
Solve:
solution = LinearSolve[stiffness, load];
Generate an Interpolating function:
mesh = methodData["ElementMesh"];
ifun = ElementMeshInterpolation[{mesh}, solution]
And visualize:
Plot3D[ifun[x, y], {x, y} [Element] mesh]
Hope this helps a bit.
$endgroup$
add a comment |
$begingroup$
I am not a 100% sure I understand your question, here is what I think could help you.
The finite element method is, strictly speaking, not a method to solve PDEs. What it does it takes a continuous PDE and converts it to an approximate equivalent matrix and vector. The matrix and vector are discrete representations of the PDE. When you solve this set of equations you will get an approximate solution to the PDE.
We set up an equation:
Needs["NDSolve`FEM`"]
{state} =
NDSolve`ProcessEquations[{Laplacian[u[x, y], {x, y}] == 1,
DirichletCondition[u[x, y] == 0, True]}, u, {x, 0, 1}, {y, 0, 1},
Method -> {"FiniteElement"}];
Now we extract some data from the NDSolve state data object.
femdata = state["FiniteElementData"]
femdata["Properties"]
methodData = femdata["FEMMethodData"];
bcData = femdata["BoundaryConditionData"];
pdeData = femdata["PDECoefficientData"];
variableData = state["VariableData"];
solutionData = state["SolutionData"][[1]];
If you do not want to / can not use NDSolve`ProcessEquations you may need to look at InitializePDECoefficients
and such functions.
If you look at pdeData, that now contains the coefficients of the equations given in NDSolve:
pdeData["All"]
{{{{1}}, {{{{0}, {0}}}}}, {{{{{-1,
0}, {0, -1}}}}, {{{{0}, {0}}}}, {{{{0,
0}}}}, {{0}}}, {{{0}}}, {{{0}}}}
Now, the finite element method is applied. This converts the continuous PDE into a system of discrete matrices:
discretePDE = DiscretizePDE[pdeData, methodData, solutionData]
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"]
You can look at the matrices:
MatrixPlot[stiffness]
The same conversion is done for the boundary conditions:
discreteBCs =
DiscretizeBoundaryConditions[bcData, methodData, solutionData];
All the finite element method does is done now. The rest is linear algebra.
We now put the discrete boundary conditions into the discrete matrices:
DeployBoundaryConditions[{load, stiffness}, discreteBCs]
Solve:
solution = LinearSolve[stiffness, load];
Generate an Interpolating function:
mesh = methodData["ElementMesh"];
ifun = ElementMeshInterpolation[{mesh}, solution]
And visualize:
Plot3D[ifun[x, y], {x, y} [Element] mesh]
Hope this helps a bit.
$endgroup$
add a comment |
$begingroup$
I am not a 100% sure I understand your question, here is what I think could help you.
The finite element method is, strictly speaking, not a method to solve PDEs. What it does it takes a continuous PDE and converts it to an approximate equivalent matrix and vector. The matrix and vector are discrete representations of the PDE. When you solve this set of equations you will get an approximate solution to the PDE.
We set up an equation:
Needs["NDSolve`FEM`"]
{state} =
NDSolve`ProcessEquations[{Laplacian[u[x, y], {x, y}] == 1,
DirichletCondition[u[x, y] == 0, True]}, u, {x, 0, 1}, {y, 0, 1},
Method -> {"FiniteElement"}];
Now we extract some data from the NDSolve state data object.
femdata = state["FiniteElementData"]
femdata["Properties"]
methodData = femdata["FEMMethodData"];
bcData = femdata["BoundaryConditionData"];
pdeData = femdata["PDECoefficientData"];
variableData = state["VariableData"];
solutionData = state["SolutionData"][[1]];
If you do not want to / can not use NDSolve`ProcessEquations you may need to look at InitializePDECoefficients
and such functions.
If you look at pdeData, that now contains the coefficients of the equations given in NDSolve:
pdeData["All"]
{{{{1}}, {{{{0}, {0}}}}}, {{{{{-1,
0}, {0, -1}}}}, {{{{0}, {0}}}}, {{{{0,
0}}}}, {{0}}}, {{{0}}}, {{{0}}}}
Now, the finite element method is applied. This converts the continuous PDE into a system of discrete matrices:
discretePDE = DiscretizePDE[pdeData, methodData, solutionData]
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"]
You can look at the matrices:
MatrixPlot[stiffness]
The same conversion is done for the boundary conditions:
discreteBCs =
DiscretizeBoundaryConditions[bcData, methodData, solutionData];
All the finite element method does is done now. The rest is linear algebra.
We now put the discrete boundary conditions into the discrete matrices:
DeployBoundaryConditions[{load, stiffness}, discreteBCs]
Solve:
solution = LinearSolve[stiffness, load];
Generate an Interpolating function:
mesh = methodData["ElementMesh"];
ifun = ElementMeshInterpolation[{mesh}, solution]
And visualize:
Plot3D[ifun[x, y], {x, y} [Element] mesh]
Hope this helps a bit.
$endgroup$
I am not a 100% sure I understand your question, here is what I think could help you.
The finite element method is, strictly speaking, not a method to solve PDEs. What it does it takes a continuous PDE and converts it to an approximate equivalent matrix and vector. The matrix and vector are discrete representations of the PDE. When you solve this set of equations you will get an approximate solution to the PDE.
We set up an equation:
Needs["NDSolve`FEM`"]
{state} =
NDSolve`ProcessEquations[{Laplacian[u[x, y], {x, y}] == 1,
DirichletCondition[u[x, y] == 0, True]}, u, {x, 0, 1}, {y, 0, 1},
Method -> {"FiniteElement"}];
Now we extract some data from the NDSolve state data object.
femdata = state["FiniteElementData"]
femdata["Properties"]
methodData = femdata["FEMMethodData"];
bcData = femdata["BoundaryConditionData"];
pdeData = femdata["PDECoefficientData"];
variableData = state["VariableData"];
solutionData = state["SolutionData"][[1]];
If you do not want to / can not use NDSolve`ProcessEquations you may need to look at InitializePDECoefficients
and such functions.
If you look at pdeData, that now contains the coefficients of the equations given in NDSolve:
pdeData["All"]
{{{{1}}, {{{{0}, {0}}}}}, {{{{{-1,
0}, {0, -1}}}}, {{{{0}, {0}}}}, {{{{0,
0}}}}, {{0}}}, {{{0}}}, {{{0}}}}
Now, the finite element method is applied. This converts the continuous PDE into a system of discrete matrices:
discretePDE = DiscretizePDE[pdeData, methodData, solutionData]
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"]
You can look at the matrices:
MatrixPlot[stiffness]
The same conversion is done for the boundary conditions:
discreteBCs =
DiscretizeBoundaryConditions[bcData, methodData, solutionData];
All the finite element method does is done now. The rest is linear algebra.
We now put the discrete boundary conditions into the discrete matrices:
DeployBoundaryConditions[{load, stiffness}, discreteBCs]
Solve:
solution = LinearSolve[stiffness, load];
Generate an Interpolating function:
mesh = methodData["ElementMesh"];
ifun = ElementMeshInterpolation[{mesh}, solution]
And visualize:
Plot3D[ifun[x, y], {x, y} [Element] mesh]
Hope this helps a bit.
edited 6 hours ago
answered 7 hours ago
user21user21
19.7k44983
19.7k44983
add a comment |
add a comment |
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$begingroup$
In what sense is this a differential equation? Normally you are looking for an unknown function that is defined through a differential equation. What is the unknown function?
$endgroup$
– Hugh
8 hours ago
$begingroup$
my code was wrong.at first,I did used Cos(xy) for creating PDE and solve by FEM. the unknown function is u[x,y]
$endgroup$
– Xminer
8 hours ago
$begingroup$
Do you have a region on which you wish to solve the equation and boundary conditions?
$endgroup$
– Hugh
8 hours ago
$begingroup$
region is {x,-1,1} and{y,-1,1}. Boundary Condition is u[-1,y]=Cos[-y]
$endgroup$
– Xminer
8 hours ago
1
$begingroup$
Can you specify what problem you encountered? Otherwise there is an entire tutorial on FiniteElementProgramming.
$endgroup$
– user21
8 hours ago