未來展望

我們所提出的低頻高頻拆解方法是透過靜止狀態的自然頻率進行拆解。假如可以提出一 個在執行時間有效率拆解方法。將會使 IMEXv 使用範圍更加廣泛具有更大吸引力。

目前我們將 IMEXv 套用在簡化的 StVK 有限元素法之上。其實 IMEXv 也可以使用 在更多物理模擬系統像是布料、流體、煙模擬之上，但需要一個有效區分出低頻位能和 高頻位能的演算法進而使用 IMEXv，進而有效保持低頻運動和高頻運動而且有效的保 持整體能量。

50

(a) (b)

(c) (d)

(e) (f)

圖 20 IMEXv 和 NEWMARK 動畫比較圖。此為馬的模型受到長達 0.3 秒施力的動畫圖。

左邊(a)、(c)、(e)是 IMEX 的 0、3、16 秒的動畫圖；右圖(b)、(d)、(f)是 NEWMARK 的相同時間動畫圖。可以發現此二積分器的結果是近乎相同的。

51

表 3 實驗一 IMEXv 所達成的模擬時間長度(秒)。在 36 組實驗裡面有 IMEXv 三組未能 夠達成設定的 30 秒的實驗。如表中的綠色部分即是無法完成的組別。

圖 21 beam 無阻尼週期性施力 IMEXv 能量圖。此為實驗一的配置，但 IMEXv 並沒有辦 法完成 30 秒的模擬，因為在 6.5 秒之後能量趨近於不穩定導致非線性求解無法有 效完成。

52

參考資料

[1] A. Stern and E. Grinspun, "Implicit-Explicit Variational Integration of Highly Oscillatory Problems," Multiscale Model. Simulation, vol. 7, pp. 1779-1794, 2009.

[2] J. Barbic and D. L. James, "Real-Time Subspace Integration for St. Venant-Kirchhoff Deformable Models," ACM Transactions on Graphics (SIGGRAPH 2005), vol. 24, pp.

982-990, August 2005.

[3] A. A. Shabana, Theory of Vibration, vol. 2: Springer-Verlag, 1991.

[4] D. Baraff and A. Witkin, "Large steps in cloth simulation," in Proceedings of the 25th annual conference on Computer graphics and interactive techniques, 1998, pp. 43-54.

[5] P. Krysl, S. Lall, and J. Marsden, "Dimensional model reduction in non linear finite element dynamics of solids and structures," International Journal for Numerical Methods in Engineering, vol. 51, pp. 479-504, 2001.

[6] M. G. Choi and H. S. Ko, "Modal warping: Real-time simulation of large rotational deformation and manipulation," IEEE Transactions on Visualization and Computer Graphics, pp. 91-101, 2005.

[7] D. L. James and K. Fatahalian, "Precomputing interactive dynamic deformable scenes," ACM Transactions on Graphics, vol. 22, pp. 879-887, 2003.

[8] L. Kharevych, W. Yang, Y. Tong, E. Kanso, J. E. Marsden, P. Schr\, \#246, der, and M.

Desbrun, "Geometric, variational integrators for computer animation," presented at the Proceedings of the 2006 ACM SIGGRAPH/Eurographics symposium on Computer animation, Vienna, Austria, 2006.

[9] H. Goldstein, C. Poole, J. Safko, and S. R. Addison, Classical mechanics vol. 70, 2002.

[10] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes vol. 3: Cambridge Univ Press, 2007.

[11] R. Bridson, S. Marino, and R. Fedkiw, "Simulation of clothing with folds and wrinkles," in ACM SIGGRAPH 2005 Courses, Los Angeles, California, 2003.

[12] G. Irving, J. Teran, and R. Fedkiw, "Invertible finite elements for robust simulation of

53

large deformation," Proceedings of the 2004 ACM SIGGRAPH/Eurographics symposium on Computer animation, pp. 131-140, 2004.

[13] E. Sifakis, T. Shinar, G. Irving, and R. Fedkiw, "Hybrid simulation of deformable solids," in Proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on Computer animation, San Diego, California, 2007, pp. 81-90.

[14] A. Selle, M. Lentine, and R. Fedkiw, "A mass spring model for hair simulation," ACM Transactions on Graphics vol. 27, pp. 1-11, 2008.

[15] B. Thomaszewski, S. Pabst, and W. Strasser, "Asynchronous cloth simulation," in Computer Graphics International, 2008.

[16] C. Kane, J. E. Marsden, M. Ortiz, and M. West, "Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems,"

International Journal for Numerical Methods in Engineering, vol. 49, pp. 1295--1325, 1999.

[17] A. Stern and M. Desbrun, "Discrete geometric mechanics for variational time integrators," presented at the ACM SIGGRAPH 2006 Courses, Boston, Massachusetts, 2006.

[18] A. Lew, J. E. Marsden, M. Ortiz, and M. West, "Asynchronous variational integrators," Archive for Rational Mechanics and Analysis, vol. 167, pp. 85-146, 2003.

[19] D. Harmon, E. Vouga, B. Smith, R. Tamstorf, and E. Grinspun, "Asynchronous contact mechanics," ACM Transactions on Graphics, vol. 28, pp. 1-12, 2009.

[20] E. Boxerman and U. Ascher, "Decomposing cloth," presented at the Proceedings of the 2004 ACM SIGGRAPH/Eurographics symposium on Computer animation, Grenoble, France, 2004.

[21] A. Stern and M. Desbrun, "Discrete geometric mechanics for variational time integrators," in Siggraph Course, 2006, pp. 1-6.

[22] R. Courant, K. Friedrichs, and H. Lewy, "On the partial difference equations of mathematical physics," IBM Journal of Research and Development, vol. 11, pp.

215-234, 1967.

[23] M. West, "Variational integrators," Ph.D. Thesis, California Institute of Technology, 2004.

[24] J. Barbic. (2009). Computer graphics research code, http://www.jernejbarbic.com/code.

54

[25] R. D. Skeel, "Variable step size destabilizes the Stormer/leapfrog/Verlet method," BIT Numerical Mathematics, vol. 33, pp. 172-175, 1993.

55

56

附錄一 小定理

變分小定理

假設 且

則 。

證明:

令

令

因為，

x

r(x)

a b

h(x)

57

附錄二 能量圖比較

以下附錄是每一個可形變模型搭配不同施力和阻尼力的能量圖，實驗條件如 5.1 所設 定。同一個表格包含四張圖能量圖，左上是無阻尼 IMEXv、右上是無阻尼 NEWMARK、

左下是具阻尼 IMEXv、右下是具阻尼 NEWMARK:

圖 22 beam 衝擊力能量圖

58

圖 23 beam 固定時間施力能量圖

59

圖 24 beam 週期性施力能量圖

60

圖 25 basket 衝擊力能量圖

61

圖 26 basket 固定時間施力能量圖

62

圖 27 basket 週期性施力能量圖

63

圖 28 bridge 固定時間施力能量圖

64

圖 29 basket 週期性施力能量圖

65

圖 30 tower 衝擊力能量圖

66

圖 31 tower 固定時間施力能量圖

67

圖 32 tower 週期性施力能量圖

68

圖 33 heart 衝擊力能量圖

69

圖 34 heart 固定時間施力能量圖

70

圖 35 heart 週期性施力能量圖

71

72

附錄三 效能數據

以下是 IMEXv 和 NEWMARK 效能比較的詳細數據:

表 4 IMEXv vs NEWMARK 有阻尼加速比較表

73

表 5 無阻尼衝擊力，效能比較表

表 6 無阻尼固定時間施力，效能比較表

74

表 7 無阻尼週期性施力，效能比較表

表 8 具阻尼衝擊力，效能比較表

75

表 9 具阻尼固定時間施力，效能比較表

表 10 具阻尼週期性施力，效能比較表