Modified moving least squares with polynomial bases for scattered data approximation. (English) Zbl 1410.65019
Summary: One common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes. To tackle such problems, we present a modified, robust moving least squares (MLS) method for scattered data smoothing and approximation. The error functional used in the derivation of the classical MLS approximation is augmented with additional terms based on the coefficients of the polynomial base functions. This allows quadratic base functions to be used with the same size of the support domain as linear base functions, resulting in better approximation capability. The increased robustness of the modified MLS method to irregular nodal distributions makes it suitable for use across many fields. The analysis is supported by several univariate and bivariate examples.
Keywords:
moving least squares; random point distribution; scattered data approximation; robust shape function generationSoftware:
Mfree2DReferences:
| [1] | Li, S.; Liu, W. K., Meshfree Particle Methods (2004), Springer- Verlag: Springer- Verlag Berlin · Zbl 1073.65002 |
| [2] | Liu, G. R., Mesh Free Methods: Moving Beyond the Finite Element Method (2003), CRC Press: CRC Press Boca Raton · Zbl 1031.74001 |
| [3] | Jin, X.; Joldes, G. R.; Miller, K.; Yang, K. H.; Wittek, A., Meshless algorithm for soft tissue cutting in surgical simulation, Comput. Methods Biomech. Biomed. Eng., 17, 800-811 (2014) |
| [4] | Zhang, G.; Wittek, A.; Joldes, G. R.; Jin, X.; Miller, K., A three-dimensional nonlinear meshfree algorithm for simulating mechanical responses of soft tissue, Eng. Anal. Bound. Elem., 42, 60-66 (2014) · Zbl 1297.74076 |
| [5] | Miller, K.; Horton, A.; Joldes, G. R.; Wittek, A., Beyond finite elements: a comprehensive, patient-specific neurosurgical simulation utilizing a meshless method, J. Biomech., 45, 2698-2701 (2012) |
| [6] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005 |
| [7] | Alexa, M.; Behr, J.; Cohen-Or, D.; Fleishman, S.; Levin, D.; Silva, C. T., Computing and rendering point set surfaces, IEEE Trans. Vis. Comput. Graph., 9, 3-15 (2003) |
| [8] | Shen, C.; O’Brien, J. F.; Shewchuk, J. R., Interpolating and approximating implicit surfaces from polygon soup, ACM Trans. Graph., 23, 896-904 (2004) |
| [9] | Muller, M.; Keiser, R.; Nealen, A.; Pauly, M.; Gross, M.; Alexa, M., Point based animation of elastic, plastic and melting objects, (Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Eurographics Association, Grenoble. Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Eurographics Association, Grenoble, France (2004)), 141-151 |
| [10] | Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Eng., 139, 3-47 (1996) · Zbl 0891.73075 |
| [11] | Horton, A.; Wittek, A.; Joldes, G. R.; Miller, K., A meshless total Lagrangian explicit dynamics algorithm for surgical simulation, Int. J. Numer. Methods Biomed. Eng., 26, 977-998 (2010) · Zbl 1193.92056 |
| [12] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Section 19.5. Linear regularization methods, Numerical Recipes: The Art of Scientific Computing (2007), Cambridge University Press: Cambridge University Press New York |
| [13] | Tikhonov, A. N.; Arsenin, V. Y., Solution of Ill-posed Problems (1977), Winston & Sons: Winston & Sons Washington · Zbl 0354.65028 |
| [14] | Miller, K., Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal., 1, 52-74 (1970) · Zbl 0214.14804 |
| [15] | Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B Methodol., 58, 267-288 (1996) · Zbl 0850.62538 |
| [16] | Zou, H.; Hastie, T., Regularization and variable selection via the elastic net, J. R. Stat. Soc. Ser. B Stat. Methodol., 67, 301-320 (2005) · Zbl 1069.62054 |
| [17] | Joldes, G. R.; Wittek, A.; Miller, K., Computation of intra-operative brain shift using dynamic relaxation, Comput. Methods Appl. Mech. Eng., 198, 3313-3320 (2009) · Zbl 1230.74136 |
| [18] | Joldes, G. R.; Wittek, A.; Miller, K., An adaptive dynamic relaxation method for solving nonlinear finite element problems. Application to brain shift estimation, Int. J. Numer. Methods Biomed. Eng., 27, 173-185 (2011) · Zbl 1370.65054 |
| [19] | Joldes, G. R.; Wittek, A.; Miller, K., Stable time step estimates for mesh-free particle methods, Int. J. Numer. Methods Eng., 91, 450-456 (2012) · Zbl 1253.74130 |
| [20] | Joldes, G. R.; Wittek, A.; Miller, K., Suite of finite element algorithms for accurate computation of soft tissue deformation for surgical simulation, Med. Image Anal., 13, 912-919 (2009) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
