高等結構學期末報告無網格法 ( Meshless methods)

高等結構學期末報告無網格法(Meshless methods) 報告者:蔡振鈞學號:M97520026 授課教授:郭世榮教授

Outline • Introduction • Meshless methods (MFS and image method) • Problem statements • Numerical examples • Contour plot • Conclusions

Numerical methods

Meshless methods and Mesh methods

Meshless methods • 1. MLPG (meshless local Petrov-Galerkin method) S. N. Atluri • 2. EFG (element free-Galerkin method) Belytschko • 3. RKPM (Reproducing kernel particle method) W K Liu • 4. BNM (Boundary node method) S. Muhkerjee • 5. Trefftz method Trefftz, Z. C. Li • 6. Boundary knot method W Chen • 7. Unity of partition • 8. SPM (smooth particle hydrodynamics) • 9. Local boundary integral equation method J Sladek • 10.Imaginary-part kernel method NTOU/MSV • 11.RBF (radial basis function) C S Chen • 12.Meshfree method G R Liu • 13.Boundary collocation method • 14.MFS (Method of fundamental solution) D.L. Young • 15.HP cloud • 16.NDIF (Nondimensional influence function method) S. W. Kang • 17.Moving least square method • 18.Method of finite sphere

Method of fundamental solutions (MFS) exterior problem Interior problem

Image method

Image group r=a

Problem statements Method of fundamental solutions y a=1 a=1 x 0.8 0.8 d=2h=10

… … Frozen points Problem statements Image method 2h y x

y a=1 a=1 x d=2h=10 Numerical approach to determine four coefficients

Numerical approach to determine four coefficients

Four coefficents, q(N),c1(N), c2(N) and e(N)

Contour plot an image solution (N=20 points) a solution by using the null-field BIEM(沈文成學長) (42 collocation points) a solution using the MFS (N=20 points)

Conclusions • 無網格法免去了其他數值方法需要切割元素，只需在上面佈點，計算上更簡單化，成為最近幾年來比較流行的方法。 • Image method 只要解4乘4的矩陣求解係數，而MFS卻要解N乘N的矩陣，其中N為所佈的點數，用矩陣計算量來比較，Image method 會比較好。 • Image method 所產生的Image points，可以看成是MFS的最佳佈點位置，我們可以說Image method 是MFS法的一種特例。

~Thanks for your kind attentions~

高等結構學期末報告無網格法 ( Meshless methods)