Research:

 

Research Grants and Projects:

1. Natural Science Foundation of China general project "Numerical methods for singular solutions of nonlinear problems and applications in elasticity" (Grant No. 19471001), 1995.1.-1997.12..

2. Doctoral Program of Education Ministry of China "Numerical Methods for Singular Solutions of Nonlinear Partial Differential Equations" (one of the principle researchers), 1997.1.-1999.12..

3. Natural Science Foundation of China general project "Numerical methods for singular solutions of nonlinear partial differential equations and microstructures" (Grant No. 19771002) (project director and principle researcher), 1998.1.-2000.12..

4. Doctoral Program of Education Ministry of China "Numerical Methods for Partial Differential Equations in Material Sciences" (one of the principle researchers), 2000.1.-2002.12..

5. The State Key Project of Basic Researches (973 projects): "Large Scale Scientific Computing Research" (Project No. G1999032800), "Multi-scale computing research on material properties" research group (G1999032802)(group leader and principle researcher), 1999.10-2001.6; "Fundamental numerical methods, Innovation and Development" research group (G1999032804)(group leader and principle researcher), 2001.7.-2004.7.

6. Doctoral Program of Education Ministry of China "Numerical Methods for Multiscale Nonlinear Partial Differential Equations" (one of the principle researchers), 2003.1.-2005.12..

7. Doctoral Program of Education Ministry of China "Nonlinear Multiscale Computational Methods for Materials Microstructure" (Project leader and principle researcher), 2004.1.-2006.12..

8. Natural Science Foundation of China major project "Adaptive Grid in Numerical Solutions of Partial Differential Equations" (Grant No.  10431050) (one of the principle researchers), 2005.1.-2008.12.

9. Natural Science Foundation of China general project "Numerical Methods for Partial differential Equations in Materials Sciences" (Grant No.  10571006) (project director and principle researcher), 2006.1.-2008.12.

10. Natural Science Foundation of China overseas youth cooperation project "Numerical Computation of Partial Differential Equations" (Grant No.  10528102), 2006.1.-2008.12. 

11.Doctoral Program of Education Ministry of China "Theory and Applications of Finite Element Adaptive Methods" (Grant No.  20060001007) (project director and principle researcher), 2007.1.-2009.12.

12. The State Key Project of Basic Researches (973 projects): "High Performance Scientific Computing Research" (Project No. 2005CB321701), 2005.12.-2010.11.

13.Natural Science Foundation of China general project "Numerical study on some nonlinear phenomenon in solids and thin films" (Grant No. 10871011) (project director and principle researcher), 2009.1.-2011.12.

14.Doctoral Program of Education Ministry of China "Numerical study on the mechanism of cavitation and fracture development" (Grant No.  20100001110004) (project director and principle researcher), 2011.1.-2013.12.

15.Natural Science Foundation of China general project "Numerical Study on Nonlinear Soft Elastic Materials" (Grant No. 11171008) (project director and principle researcher), 2012.1.-2015.12.

 

16.Natural Science Foundation of China general project " Numerical study on cavitation in incompressible nonlinear elasticity " (Grant No. 11571022) (project director and principle researcher), 2016.1.-2019.12.

 

 

Main Research Fields and Results:

1. Numerical methods for singular solutions with Lavrentiev phenomenon.

1926, Lavrentiev found that standard numerical methods always fail to approximate  the singular solutions of some variational problems. 1987, J. M. Ball and G. Knowles successfully established a numerical method which can overcome the Lavrentiev phenomenon.

I introduced other two numerical methods (element removal method and finite element  truncation method), proved their convergence and applied them to elastic problems (see Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon,  Proc. R. Soc. Lond., 439A1992, pp.131-137;  A numerical method for computing singular minimizers.  Numerische Mathematik, 71(1995), pp.317-330; and Existence of minimizers and  microstructure in nonlinear elasticity. Nonlinear Analysis, 27(3)1996, pp.297-308.). For further development of the truncation method (convergence under weaker conditions, simpler implementation techniques, numerical estimation of singularities, computation of continuous Sobolev index dependent Lavrentiev gap and elastic problem with Lavrentiev phenomenonsee (Yu Bai and Zhiping Li) A truncation method for detecting singular minimizers involving the Lavrentiev phenomenon, Math. Models Methods Appl. Sci., 1 6(6)2006, pp.847-867, and (Yu Bai and Zhiping Li) Numerical Solution of nonlinear elasticity problems with Lavrentiev phenomenon, Math. Models Methods Appl. Sci., 17(10)2007, pp.1619-1640. 

 

2. Lower semi-continuity of integral functionals and integral representation of lower semicontinuous envelopes of integral functionals

The new point of my work (see: A theorem on lower semicontinuity of integral functionalsProc. R. Soc. Edinburgh, 126A(1996), pp.363-374Lower semicontinuity of multiple integrals and convergent integrands. ESAIM: Control, Optimisation and Calculus of Variations 1(1996), pp.169-189.); and An integral representation theorem for lower semicontinuous envelopes of integral functionalsNonliear Analysis, 32(4)1998, pp.541-548.is that the convexity (or quasi-convexity) condition is not imposed on the sequence of the integral functionals but instead on the limit functional only, and this makes it convenient to apply the theory to the numerical analysis.

3. Mathematical theory and numerical method for crystalline microstructure

(a) A coupled numerical method is established to compute the relaxed minimizer and the microstructure simultaneously (see Simultaneous numerical approximation of microstructures and relaxed minimizers. Numerische Mathematik, 78(1)(1997), pp. 21-38). The result is mainly of theoretical value.  A related extension of the work is about a numerical method for computing the lower semicontinuous envelope of non-convex integral functionals (see  Computation of Lower Semicontinuous Envelope of Integral Functionals and Non-homogeneous MicrostructuresNonlinear Analysis68 (2008)2058-2071).

(b) A series of numerical methods for computing crystalline microstructures (see Rotational transformation method and some numerical techniques for computing microstructuresMath. Models Meth. Appl. Sci., 8(6)1998, pp. 985-1002);  A mesh transformation method for computing microstructures,   Numer. Math., 89(3)(2001) ; and A Periodic relaxation method for computing microstructures,  Appl. Numer. Math., 32(2000), pp. 291-303). The methods dramatically increase the efficiency of the computation, especially for the simple laminated microstructures, and provide a powerful computing tool for further study of more complicated microstructures. The mesh transformation method is capable of capturing weak discontinuities of the minimizers of variational problems with high accuracy (see (Jiansong Zhou, Zhiping Li) Computing Non-smooth minimizers with the mesh transformation methodIMA J. Numer. Anal., 252005, 458-472.

(c) An example is given to show that a non-conforming finite element method systematically produces a pseudo-microstructure (see Laminated microstructure in a variational problems with a non-rank-one connected double well potential  J. Math. Anal. Appl., 217(1998), pp.490-500).

(d) A numerical method is established to compute the finite order laminates in laminates (see Finite order rank-one convex envelopes and computation of microstructures with laminates in laminatesBIT Numer. Math., 40(2000), 745-761). The method is further developed into a highly efficient iterative method for computing the finite order rank-one convex envelopes (see (Xin Wang, Zhiping Li) A numerical iterative scheme for computing finite order rank-one convex envelopesAppl. Math. Comp., 182007, 19-30.

(e) Established a mesh transformation and regularization method, and successfully computed the needle-like microstructures and quasi-static simulation of the growth of twin microstructures (see  Computational Materials Science 21(2001), pp. 418-428and Appl. Numer. Math., 39(2001), pp. 1-15).

(f) Applications have been made in the computation of needle-like microstructures, branched laminated microstructures and the corresponding scaling laws (see  A numerical study on the Scale of Laminated Microstructure with Surface Energy.  Material Sci. & Engrg, A343(2003),182-193 Numerical Justification of Branched Laminated Microstructure with Surface Energy. SIAM J. Sci. Compt.,24(3)(2003),1054-1075. (Liying Liu, Zhiping Li) Computation of length scales for second-order laminated microstructure with surface energyAppl. Math. Modelling, 312007, 245-258.)

(g) Applications have also been made in the computation of stress induced microstructure, multiscale modeling and  computation of microstructures (see Numerical computation of stress induced microstructure, Science in China, series A, vol. 47, supp.,2004, pp. 165-171, and Multiscale modelling and computation of microstructures in multi-well problems, Math. Mod. Meth. Appl. Sci., 9(14)2004, 1343-1360.).

 

4. Numerical Analysis and Computation of Micromagnetics

A numerical method using  multi-atomic Young measure and artificial boundary is established for approximation of micromagnetics (see  Multi-atomic Young measure and artificial boundary in approximation of micromagnetics, Appl. Numer. Math., 51(1)2004, 69-88 (a joint work with Xiaonan Wu)). 

Established an efficient algorithm using non-conforming FEM combined with the multi-atomic Young measure and artificial boundary (see (Xianmin Xu and Zhiping Li) Non-Conforming Finite Element and Artificial Boundary in Multi-atomic Young Measure Approximation for MicromagneticsAppl. Numer. Math., 59 (2009), 920-937), proved the convergence and stability of the method (see (Zhiping Li and Xianmin Xu) Convergence and Stability of a Numerical Method for MicromagneticsNumerische Mathematik112(2009), 245-265), derived a reliable postriori error estimator and an adaptive algorithm (see (Xianmin Xu and Zhiping Li) A Posteriori Error Estimates of a Non-conforming Finite Element Method for Problems with Artificial Boundary Conditions, J. Comp. Math., 27(6) 2009, 677-696. (Xianmin Xu and Zhiping Li) (DOI: 10.4208//jcm.2009.09-m2608)).

 

5. Mathematical Modeling and Computation of Buckling and Wrinkling of Elastic Films

By means of mathematical modeling and numerical simulation, the wrinkling initiation, evolution and pattern formation process of a compressed anisotropic elastic thin film on a viscous layer is studied (see (Wei Jiang and Zhiping Li) A Numerical Study of Wrinkling Evolution of an Elastic Film on a Viscous LayerModelling Simul. Mater. Sci. Eng., 17 (2009), 055010.

An annular sector model for the telephone cord buckles of elastic thin films on rigid substrates is established, which successfully simulated the telephone cord buckles with two humps along the ridge of each section of a buckle (see (Shan Wang and Zhiping Li) Mathematical modeling and numerical simulation of telephone cord buckles of elastic films, Sci. China Math., 54(5) 2011: 1063-1076.).

 

The morphology of telephone cord buckles of elastic thin films, obtained by an annular sector model established using the von Karman plate equations in polar coordinates, can be used to evaluate the initial residual stress and interface toughness of the filmsubstrate system (see (Shan Wang and Zhiping Li) Evaluation of mechanical parameters of an elastic thin film system by modeling and numerical simulation of telephone cord buckles, J. Comput. Appl. Math.,236 (5) 2011, 860866.).

 

A new collocation method with multiple-endpoints and a new boundary condition technique is established for high order differential equations. An example on nonlinear elastic thin film buckling shows the advantage of the new method for high order nonlinear partial differential equations with complex boundary conditions (see (Shan Wang and Zhiping Li) A Multiple-endpoints Chebysheve Collocation Method for High Order Differential EquationsContemporary Mathematics, Volume 586 (2013), 365-373.).

 

6. Numerical Computation and Analysis for Cavitation Problems in Nonlinear elasticity

A dual-parametric finite element method is introduced for the computation of singular minimizers in the 2D cavitation problem in nonlinear elasticity. The method overcomes the difficulties, such as the mesh entanglement and material interpenetration, generally encountered in the finite element approximation of problems with extremely large expansionary deformation see (Yijiang Lian and Zhiping Li) A dual-parametric finite element method for cavitation in nonlinear elasticity, J. Comput. Appl. Math., 236 (5) 2011, 834842..

 

An iso-parametric finite element method is introduced to study cavitations and configurational forces in nonlinear elasticity. The method is shown to be highly efficient in capturing the cavitation phenomenon, especially in dealing with multiple cavities of various sizes and shapes. Numerical experiments verified and extended, for a class of nonlinear elasticity materials, the theory of Sivaloganathan and Spector on the configurational forces of cavities, as well as justified a crucial hypothesis of the theory on the cavities (see (Yijiang Lian and Zhiping Li) A Numerical Study on Cavitation in Nonlinear Elasticity ---- Defects and Configurational Forces, Math. Mod. Meth. Appl. Sci., 21(12)2011, 25512574.). For certain compressible hyper-elastic material, our numerical experiments on the case of two voids revealed that both the positions and initial sizes of the pre-existing voids can have significant effects on the final grown configuration (see (Yijiang Lian and Zhiping Li) Position and size effects on voids growth in nonlinear elasticityInt. Journal of Fracture, 173(2), pp 147-161, 2012).

 

The orientation-preservation conditions and approximation errors of a dual-parametric bi-quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed. The analytical results allow us to establish, based on an error equidistribution principle, an optimal meshing strategy for the method in cavitation computation (see (Chunmei Su and Zhiping Li) Error Analysis of a Dual-parametric Bi-quadratic FEM in Cavitation Computation in Elasticity. SIAM Journal on Numerical Analysis. Vol. 53, No. 3. July 2015. pp. 1629-1649.).

 

A set of necessary and sufficient conditions are derived for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations (see (Chunmei Su and Zhiping Li) Orientation-Preservation Conditions on an Iso-parametric FEM in Cavitation ComputationScience in China: Mathematics40(4) 2017719-734.).

 

The approximation properties of the quadratic iso-parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed. More precisely, (1) the finite element interpolation errors are established in terms of the mesh parameters; (2) a mesh distribution strategy based on an error equi-distribution principle is given; (3) the convergence of finite element cavity solutions is proved. Numerical experiments show that, in fact, the optimal convergence rate can be achieved by the numerical cavity solutions (see (Chunmei Su and Zhiping Li) A meshing strategy for a quadratic iso-parametric FEM incavitation computation in nonlinear elasticity, Journal of Computational and Applied Mathematics 330 (2018) 630647.).

 

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved (see (Liang Wei and Zhiping Li) Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity, Front. Math. China, 13(1) 2018: 203226.).

 

 

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