Mapping a curve-sided quadrilateral to a rectangle












2














I am currently investigating different ways of solving the Laplace equation
$$frac{partial^2 F}{partial x^2} + frac{partial^2 F}{partial z^2} = 0 $$
numerically on the domain $Omega$ shown as the shaded region in the figure below. As the figure shows, $Omega$ is bounded by the vertical lines $x = 0$, $x = L$, the horizontal line $z = -h$ and the curve $(x, eta(x))$ where $eta(x)$ is a smooth, $L$-periodic function.



One strategy for solving the Laplace equation is to make a coordinate transformation $(x,z) mapsto (r(x,z), s(x,z))$ and then solve the equation that the function $G(r(x,z), s(x,z)) = F(x,z)$ satisfies on the transformed domain. For this strategy to be practical, the coordinate transformation should map $Omega$ to a rectangle.



Now, my questions (which is probably more of a request) is the following: Which coordinate tranformations from $Omega$ to a rectangle exist?
I am aware of the simple vertical stretching $(x,z) mapsto (x, s(x,z))$ where
$$ s(x,z) = frac{2z + h - eta(x)}{h + eta(x)}$$
and in principal it works fine. In the mapped coordinates the function $G(x,s(x,z)) = F(x,z)$ solves the equation
$$ frac{partial^2 G}{partial x^2} + bigg( bigg[frac{partial s}{partial x} bigg]^2 + bigg[ frac{partial s}{partial z} bigg]^2 bigg) frac{partial^2 G}{partial s^2} + 2 frac{partial s}{partial x} frac{partial^2 G}{partial x partial s} + frac{partial^2 s}{partial x^2} frac{partial G}{partial s} = 0, qquad 0 leq x leq L, , , , -1 leq s leq 1$$
and while this certainly can be solved numerically, it would be nice to have a coordinate transformation which does not give rise to such a complicated equation. The ideal transformation would of course be a conformal one, since the Laplace equation is invariant under such a change of coordinates.



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    2














    I am currently investigating different ways of solving the Laplace equation
    $$frac{partial^2 F}{partial x^2} + frac{partial^2 F}{partial z^2} = 0 $$
    numerically on the domain $Omega$ shown as the shaded region in the figure below. As the figure shows, $Omega$ is bounded by the vertical lines $x = 0$, $x = L$, the horizontal line $z = -h$ and the curve $(x, eta(x))$ where $eta(x)$ is a smooth, $L$-periodic function.



    One strategy for solving the Laplace equation is to make a coordinate transformation $(x,z) mapsto (r(x,z), s(x,z))$ and then solve the equation that the function $G(r(x,z), s(x,z)) = F(x,z)$ satisfies on the transformed domain. For this strategy to be practical, the coordinate transformation should map $Omega$ to a rectangle.



    Now, my questions (which is probably more of a request) is the following: Which coordinate tranformations from $Omega$ to a rectangle exist?
    I am aware of the simple vertical stretching $(x,z) mapsto (x, s(x,z))$ where
    $$ s(x,z) = frac{2z + h - eta(x)}{h + eta(x)}$$
    and in principal it works fine. In the mapped coordinates the function $G(x,s(x,z)) = F(x,z)$ solves the equation
    $$ frac{partial^2 G}{partial x^2} + bigg( bigg[frac{partial s}{partial x} bigg]^2 + bigg[ frac{partial s}{partial z} bigg]^2 bigg) frac{partial^2 G}{partial s^2} + 2 frac{partial s}{partial x} frac{partial^2 G}{partial x partial s} + frac{partial^2 s}{partial x^2} frac{partial G}{partial s} = 0, qquad 0 leq x leq L, , , , -1 leq s leq 1$$
    and while this certainly can be solved numerically, it would be nice to have a coordinate transformation which does not give rise to such a complicated equation. The ideal transformation would of course be a conformal one, since the Laplace equation is invariant under such a change of coordinates.



    enter image description here






    differential-geometry pde harmonic-functions conformal-geometry







    share|cite|improve this question

























      2












      2








      2







      I am currently investigating different ways of solving the Laplace equation
      $$frac{partial^2 F}{partial x^2} + frac{partial^2 F}{partial z^2} = 0 $$
      numerically on the domain $Omega$ shown as the shaded region in the figure below. As the figure shows, $Omega$ is bounded by the vertical lines $x = 0$, $x = L$, the horizontal line $z = -h$ and the curve $(x, eta(x))$ where $eta(x)$ is a smooth, $L$-periodic function.



      One strategy for solving the Laplace equation is to make a coordinate transformation $(x,z) mapsto (r(x,z), s(x,z))$ and then solve the equation that the function $G(r(x,z), s(x,z)) = F(x,z)$ satisfies on the transformed domain. For this strategy to be practical, the coordinate transformation should map $Omega$ to a rectangle.



      Now, my questions (which is probably more of a request) is the following: Which coordinate tranformations from $Omega$ to a rectangle exist?
      I am aware of the simple vertical stretching $(x,z) mapsto (x, s(x,z))$ where
      $$ s(x,z) = frac{2z + h - eta(x)}{h + eta(x)}$$
      and in principal it works fine. In the mapped coordinates the function $G(x,s(x,z)) = F(x,z)$ solves the equation
      $$ frac{partial^2 G}{partial x^2} + bigg( bigg[frac{partial s}{partial x} bigg]^2 + bigg[ frac{partial s}{partial z} bigg]^2 bigg) frac{partial^2 G}{partial s^2} + 2 frac{partial s}{partial x} frac{partial^2 G}{partial x partial s} + frac{partial^2 s}{partial x^2} frac{partial G}{partial s} = 0, qquad 0 leq x leq L, , , , -1 leq s leq 1$$
      and while this certainly can be solved numerically, it would be nice to have a coordinate transformation which does not give rise to such a complicated equation. The ideal transformation would of course be a conformal one, since the Laplace equation is invariant under such a change of coordinates.



      enter image description here






      differential-geometry pde harmonic-functions conformal-geometry







      share|cite|improve this question













      I am currently investigating different ways of solving the Laplace equation
      $$frac{partial^2 F}{partial x^2} + frac{partial^2 F}{partial z^2} = 0 $$
      numerically on the domain $Omega$ shown as the shaded region in the figure below. As the figure shows, $Omega$ is bounded by the vertical lines $x = 0$, $x = L$, the horizontal line $z = -h$ and the curve $(x, eta(x))$ where $eta(x)$ is a smooth, $L$-periodic function.



      One strategy for solving the Laplace equation is to make a coordinate transformation $(x,z) mapsto (r(x,z), s(x,z))$ and then solve the equation that the function $G(r(x,z), s(x,z)) = F(x,z)$ satisfies on the transformed domain. For this strategy to be practical, the coordinate transformation should map $Omega$ to a rectangle.



      Now, my questions (which is probably more of a request) is the following: Which coordinate tranformations from $Omega$ to a rectangle exist?
      I am aware of the simple vertical stretching $(x,z) mapsto (x, s(x,z))$ where
      $$ s(x,z) = frac{2z + h - eta(x)}{h + eta(x)}$$
      and in principal it works fine. In the mapped coordinates the function $G(x,s(x,z)) = F(x,z)$ solves the equation
      $$ frac{partial^2 G}{partial x^2} + bigg( bigg[frac{partial s}{partial x} bigg]^2 + bigg[ frac{partial s}{partial z} bigg]^2 bigg) frac{partial^2 G}{partial s^2} + 2 frac{partial s}{partial x} frac{partial^2 G}{partial x partial s} + frac{partial^2 s}{partial x^2} frac{partial G}{partial s} = 0, qquad 0 leq x leq L, , , , -1 leq s leq 1$$
      and while this certainly can be solved numerically, it would be nice to have a coordinate transformation which does not give rise to such a complicated equation. The ideal transformation would of course be a conformal one, since the Laplace equation is invariant under such a change of coordinates.



      enter image description here






      differential-geometry pde harmonic-functions conformal-geometry




      differential-geometry pde harmonic-functions conformal-geometry






      share|cite|improve this question













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      Mathias Klahn

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