计算科学 - 四节点矩形元素形状矩阵 - 吾爱随笔录

我从事一个需要计算声腔模态分析的项目。空腔是刚性的，可以将问题转化为以下等式

\frac{\partial^{2} p}{\partial x^{2}} + \frac{\partial^{2} p}{\partial y^{2}} + k^{2} p = 0

$\frac{\partial^2p}{\partial{}x^2}+\frac{\partial^2p}{\partial{}y^2}+k^2p=0$

和边界条件

\partial_{x} p (y = 0) = 0, \partial_{x} p (y = L_{y}) = 0, \partial_{y} p (x = 0) = 0, \partial_{y} p (x = L_{x}) .

$\partial_xp(y=0)=0,\quad\partial_xp(y=L_y)=0,\quad\partial_yp(x=0)=0,\quad\partial_yp(x=L_x).$

我正在尝试使用有限元方法解决问题，尤其是使用四节点矩形元素。感谢这个问题的答案，我得到了表单函数

\begin{aligned} N_{1} = \frac{1}{Δ x Δ y} (x - x_{2}) (y - y_{4}) \\ N_{2} = - \frac{1}{Δ x Δ y} (x - x_{1}) (y - y_{3}), \\ N_{3} = \frac{1}{Δ x Δ y} (x - x_{4}) (y - y_{2}), \\ N_{4} = - \frac{1}{Δ x Δ y} (x - x_{3}) (y - y_{1}) . \end{aligned}

$\begin{align*} &N_1=\frac{1}{\Delta{}x\Delta{}y}(x-x_2)(y-y_4)\\ &N_2=-\frac{1}{\Delta{}x\Delta{}y}(x-x_1)(y-y_3),\\ &N_3=\frac{1}{\Delta{}x\Delta{}y}(x-x_4)(y-y_2),\\ &N_4=-\frac{1}{\Delta{}x\Delta{}y}(x-x_3)(y-y_1). \end{align*}$

现在我正在尝试计算问题的刚度和质量矩阵。为此，我使用了以下 Mathematica 脚本：

N1[x_, y_] := (x - x2) (y - y4)/(x1 - x2)/(y1 - y4)
N2[x_, y_] := (x - x1) (y - y3)/(x2 - x1)/(y2 - y3)
N3[x_, y_] := (x - x4) (y - y2)/(x3 - x4)/(y3 - y4)
N4[x_, y_] := (x - x3) (y - y1)/(x4 - x3)/(y4 - y1)
DxN1[x_, y_] := D[N1[x, y], x]
DxN2[x_, y_] := D[N2[x, y], x]
DxN3[x_, y_] := D[N3[x, y], x]
DxN4[x_, y_] := D[N4[x, y], x]
DyN1[x_, y_] := D[N1[x, y], y]
DyN2[x_, y_] := D[N2[x, y], y]
DyN3[x_, y_] := D[N3[x, y], y]
DyN4[x_, y_] := D[N4[x, y], y]
Kx11[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN1[x, y]*DxN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx12[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN1[x, y]*DxN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx13[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN1[x, y]*DxN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx14[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN1[x, y]*DxN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx21[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN2[x, y]*DxN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx22[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN2[x, y]*DxN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx23[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN2[x, y]*DxN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx24[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN2[x, y]*DxN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx31[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN3[x, y]*DxN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx32[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN3[x, y]*DxN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx33[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN3[x, y]*DxN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx34[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN3[x, y]*DxN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx41[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN4[x, y]*DxN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx42[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN4[x, y]*DxN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx43[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN4[x, y]*DxN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Kx44[x_, y_] := 
 Simplify[Integrate[
   Integrate[DxN4[x, y]*DxN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky11[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN1[x, y]*DyN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky12[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN1[x, y]*DyN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky13[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN1[x, y]*DyN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky14[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN1[x, y]*DyN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky21[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN2[x, y]*DyN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky22[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN2[x, y]*DyN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky23[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN2[x, y]*DyN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky24[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN2[x, y]*DyN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky31[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN3[x, y]*DyN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky32[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN3[x, y]*DyN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky33[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN3[x, y]*DyN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky34[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN3[x, y]*DyN4[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky41[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN4[x, y]*DyN1[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky42[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN4[x, y]*DyN2[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky43[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN4[x, y]*DyN3[x, y], {x, x1, x2}], {y, y1, y2}]]
Ky44[x_, y_] := 
 Simplify[Integrate[
   Integrate[DyN4[x, y]*DyN4[x, y], {x, x1, x2}], {y, y1, y2}]]
{{Kx11[x, y], Kx12[x, y], Kx13[x, y], Kx14[x, y]}, {Kx21[x, y], 
   Kx22[x, y], Kx23[x, y], 
   Kx24[x, y]}, {Kx31[x, y], Kx32[x, y], Kx33[x, y], 
    Kx34[x, y]} {Kx41[x, y], Kx42[x, y], Kx43[x, y], 
    Kx44[x, y]}} // MatrixForm
{{Ky11[x, y], Ky12[x, y], Ky13[x, y], Ky14[x, y]}, {Ky21[x, y], 
   Ky22[x, y], Ky23[x, y], 
   Ky24[x, y]}, {Ky31[x, y], Ky32[x, y], Ky33[x, y], 
    Ky34[x, y]} {Ky41[x, y], Ky42[x, y], Ky43[x, y], 
    Ky44[x, y]}} // MatrixForm

我不喜欢输出，因为它不像以前那样对称：

有没有人表达我的问题的刚度和质量矩阵？

问候。