​Biomechanics is the study of the structure, function, and motion of the mechanical aspects of biological systems. From the level of continuum mechanics, two problems are of high interest. One is to identify proper growth tensors to regenerate the morphologies and the stress state of the biological tissues or organs (such as the simulation of the bitter gourd above); the other is to relate the growth tensors to biological information, such as the genes and morphogens. The results in this field can give explanations for various growth phenomena in the view of continuum mechanics, and the underlying growth principles obtained have the potential to be used in the intervention of some diseases, and designing and optimizing biomimetic structures.

Finite geometry is a branch of combinatorics. Contrary to the familiar Euclidean geometry in which a line contains infinitely many points, a finite geometry is any geometric system that has only a finite number of points. Examples include finite projective spaces and finite affine spaces, generalized polygons, finite Möbius planes and finite Laguerre planes. It has applications to groups, codes, graphs, designs and permutation polynomials, such as to extremal graph and theory, Latin squares and maximum distance separable codes.

Inverse problems are mathematical problems set up to determine the cause from the effects. Scattering problems arise from the physical phenomenon of wave–matter interaction in which the wavelength is comparable to the typical dimension of the inhomogeneity. An inverse scattering problem is to the recover the unknown/inaccessible physical objects by the measurement of wave as the result of the interaction between the object with a given input field. Applications include nano-optics, super-resolution imaging, medical imaging, nondestructive testing, geophysical exploration, radar and sonar.

The convection-dominate problem is an important problem of Computational Fluid Dynamics, which is ubiquitous in science and engineering, such as plasmas simulation in nuclear fusion, semiconductor device simulation, numerical weather prediction, etc. The numerical schemes of the convection-dominate problem have been greatly developed in both theory and applications, but there is still a lot of room of improvements in the design of numerical schemes. For instance, many existing explicit schemes are subject to the stringent time-stepping constraint; it is still a lot room of improvements in designing schemes that can capture the complex structure (such as discontinuity) accurately and robustly; it is still extremely expensive to compute some practical largely multi-scale problems.

To cure the illness in the brain glioblastoma, the Gliadel wafer, as the first FDA-approved chemotherapy, was available on the market since 1997. Due to the complex studies in vivo, the numerical methods (Finite Element Method) and tools (AFEPack) may be the effective way to simulate the dynamic process in the brain, such as the pressure in the normal brain tissue and the change of concentration of the drug release in brain tumor.