11 Meshless Algorithms for Computational Biomechanics of the Brain
299
15. Maurel, B., Combescure, A., Potapov, S.: A robust SPH formulation for solids. Eur. J. Comput.
Mech. 15, 495–512 (2006)
16. Bourantas, G.C., Mountris, K.A., Loukopoulos, V.C., Lavier, L., Joldes, G.R., Wittek, A.,
Miller, K.: Strong-form approach to elasticity: hybrid finite difference-meshless collocation
method (FDMCM). Appl. Math. Model. 57, 316–338 (2018)
17. Toma, M.: The emerging use of SPH in biomedical applications. Significances Bioeng. Biosci.
1, 1–4 (2017)
18. Zhang, Y.J., Joldes, G.R., Wittek, A., Miller, K.: Patient-specific computational biomechanics
of the brain without segmentation and meshing. Int. J. Numer. Methods Biomed. Eng. 29,
293–308 (2013)
19. Zhang, G.Y., 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, 7 (2013)
20. 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. Engin. 17, 800–817
(2014)
21. Joldes, G.R., Wittek, A., Miller, K.: Adaptive numerical integration in element-free Galerkin
methods for elliptic boundary value problems. Eng. Anal. Bound. Elem. 51, 52–63 (2015)
22. Chowdhury, H.A., Wittek, A., Miller, K., Joldes, G.R.: An element free Galerkin method based
on the modified moving least squares approximation. J. Sci. Comput. 71, 1197–1211 (2017)
23. Joldes, G.R., Chowdhury, H., Wittek, A., Miller, K.: A new method for essential boundary
conditions imposition in explicit meshless methods. Eng. Anal. Bound. Elem. 80, 94–104
(2017)
24. Joldes, G.R., Bourantas, G., Zwick, B., Chowdhury, H., Wittek, A., Agrawal, S., Mountris, K.,
Hyde, D., Warfield, S.K., Miller, K.: Suite of meshless algorithms for accurate computation of
soft tissue deformation for surgical simulation. Med. Image Anal. 56, 152–171 (2019)
25. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares methods. Math.
Comput. 37, 141–158 (1981)
26. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)
27. Joldes, G.R., Chowdhury, H.A., Wittek, A., Doyle, B., Miller, K.: Modified moving least
squares with polynomial bases for scattered data approximation. Appl. Math. Comput. 266,
893–902 (2015)
28. 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)
29. Li, M., Miller, K., Joldes, G.R., Kikinis, R., Wittek, A.: Biomechanical model for computing
deformations for whole-body image registration: a meshless approach. Int. J. Numer. Methods
Biomed. Eng. 32, e02771–e02718 (2016)
30. Heye, Z., Linwei, W., Hunter, P.J., Pengcheng, S.: Meshfree framework for image-derived
modelling. In: 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano
to Macro, pp. 1449–1452 (2008)
31. A Dassault Systèmes Simulia Corporation: SIMULIA User Assistance 2018: ABAQUS (2018),
Dassault Systemes, France
32. Chowdhury, H., Joldes, G., Wittek, A., Doyle, B., Pasternak, E., Miller, K.: Implementation of
a modified moving least squares approximation for predicting soft tissue deformation using a
meshless method. In: Doyle, B., Miller, K., Wittek, A., Nielsen, P.M.F. (eds.) Computational
Biomechanics for Medicine, pp. 59–71. Springer International Publishing, Cham (2015)
33. Joldes, G.R., Teakle, P., Wittek, A., Miller, K.: Computation of accurate solutions when using
element-free Galerkin methods for solving structural problems. Eng. Comput. 34, 902–920
(2017)
34. Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca
Raton (2003)
299
15. Maurel, B., Combescure, A., Potapov, S.: A robust SPH formulation for solids. Eur. J. Comput.
Mech. 15, 495–512 (2006)
16. Bourantas, G.C., Mountris, K.A., Loukopoulos, V.C., Lavier, L., Joldes, G.R., Wittek, A.,
Miller, K.: Strong-form approach to elasticity: hybrid finite difference-meshless collocation
method (FDMCM). Appl. Math. Model. 57, 316–338 (2018)
17. Toma, M.: The emerging use of SPH in biomedical applications. Significances Bioeng. Biosci.
1, 1–4 (2017)
18. Zhang, Y.J., Joldes, G.R., Wittek, A., Miller, K.: Patient-specific computational biomechanics
of the brain without segmentation and meshing. Int. J. Numer. Methods Biomed. Eng. 29,
293–308 (2013)
19. Zhang, G.Y., 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, 7 (2013)
20. 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. Engin. 17, 800–817
(2014)
21. Joldes, G.R., Wittek, A., Miller, K.: Adaptive numerical integration in element-free Galerkin
methods for elliptic boundary value problems. Eng. Anal. Bound. Elem. 51, 52–63 (2015)
22. Chowdhury, H.A., Wittek, A., Miller, K., Joldes, G.R.: An element free Galerkin method based
on the modified moving least squares approximation. J. Sci. Comput. 71, 1197–1211 (2017)
23. Joldes, G.R., Chowdhury, H., Wittek, A., Miller, K.: A new method for essential boundary
conditions imposition in explicit meshless methods. Eng. Anal. Bound. Elem. 80, 94–104
(2017)
24. Joldes, G.R., Bourantas, G., Zwick, B., Chowdhury, H., Wittek, A., Agrawal, S., Mountris, K.,
Hyde, D., Warfield, S.K., Miller, K.: Suite of meshless algorithms for accurate computation of
soft tissue deformation for surgical simulation. Med. Image Anal. 56, 152–171 (2019)
25. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least-squares methods. Math.
Comput. 37, 141–158 (1981)
26. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)
27. Joldes, G.R., Chowdhury, H.A., Wittek, A., Doyle, B., Miller, K.: Modified moving least
squares with polynomial bases for scattered data approximation. Appl. Math. Comput. 266,
893–902 (2015)
28. 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)
29. Li, M., Miller, K., Joldes, G.R., Kikinis, R., Wittek, A.: Biomechanical model for computing
deformations for whole-body image registration: a meshless approach. Int. J. Numer. Methods
Biomed. Eng. 32, e02771–e02718 (2016)
30. Heye, Z., Linwei, W., Hunter, P.J., Pengcheng, S.: Meshfree framework for image-derived
modelling. In: 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano
to Macro, pp. 1449–1452 (2008)
31. A Dassault Systèmes Simulia Corporation: SIMULIA User Assistance 2018: ABAQUS (2018),
Dassault Systemes, France
32. Chowdhury, H., Joldes, G., Wittek, A., Doyle, B., Pasternak, E., Miller, K.: Implementation of
a modified moving least squares approximation for predicting soft tissue deformation using a
meshless method. In: Doyle, B., Miller, K., Wittek, A., Nielsen, P.M.F. (eds.) Computational
Biomechanics for Medicine, pp. 59–71. Springer International Publishing, Cham (2015)
33. Joldes, G.R., Teakle, P., Wittek, A., Miller, K.: Computation of accurate solutions when using
element-free Galerkin methods for solving structural problems. Eng. Comput. 34, 902–920
(2017)
34. Liu, G.R.: Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, Boca
Raton (2003)