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

**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.,
439A**（1992）, pp.131-137; A numerical method
for computing singular minimizers.

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 functionals.
**Proc. R. Soc. Edinburgh, 126A**(1996),
pp.363-374；Lower semicontinuity
of multiple integrals and convergent integrands.

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
Microstructures，**Nonlinear** **Analysis****，68** (2008)，2058-2071).

(b)
A series of numerical methods for computing crystalline microstructures (see Rotational transformation method and some numerical techniques for
computing microstructures， **Math.
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 method，**IMA J. Numer.
Anal., 25****（2005）, 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 laminates, **BIT 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
envelopes，**Appl.
Math. Comp., 18****（2007）, 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. 418-428，**and** **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 energy， **Appl. Math. Modelling, 31****（2007）, 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 Micromagnetics，**Appl. 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 Micromagnetics，**Numerische**** Mathematik，112**(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 Layer， **Modelling 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 film****–substrate 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, 860–866.)**. **

**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 Equations， **Contemporary 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, 834–842.**）.
**

**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, 2551–2574.**).
**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
elasticity，**Int. 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
Computation，**Science
in China: Mathematics**，**40**(4) 2017：719-734.)**. **

**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) 630–647.)**.**

**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. **,