Investigate stretch factors

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.

[alexfikl]

Added a simple case with a warped mesh and just a few elements to see what's going on and they don't seem to be discontinuous here. Not sure what it doesn't like about the torus.

On the other hand, the triangle and square cases seem to be seeing very different values.

[alexfikl]

This is plotting the largest singular value i.e. "largest stretch factor".

Left is simplex (which has 4 elements) and right is the tensor product (which has 2 elements). They're both nice and continuous, but annoyingly different.

Interestingly, for the simplex, the sum of the two singular values does give something that vaguely matches the tensor product picture.

EDIT: To add a bit of math to this. The transformation used above is just

x = [x, y, sin(x) * sin(y)]

and the largest eigenvalue of that (according to Mathematica) looks like the tensor product plot above. Not quite sure what the simplex case is seeing :\

[alexfikl]

For future reference, a few remarks for this

• the simplex and quad cases have no right to match, since the simplex is skewed to an equilateral reference element. That means that none of the vertex angles and edge lengths of the reference elements match between the two, so the stretch factors won't match either.
• stretch factors don't have a right to be continuous, since two adjacent elements can be stretched the same along one axis, but very differently along another.
• on the gmsh meshes, there's no difference (to eye ball norms) between simplices and quads, i.e. the stretch factors are discontinuous everywhere and generally nasty looking on both.