SUMMARYAn element-free Galerkin method which is applicable to arbitrary shapes but ,requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

Bioprinting is a process based on additive manufacturing from materials containing living cells. These materials, often referred to as bioink, are based on cytocompatible hydrogel precursor formulations, which gel in a manner compatible with different bioprinting approaches. The bioink properties before, during and after gelation are essential for its printability, comprising such features as achievable structural resolution, shape fidelity and cell survival. However, it is the final properties of the matured bioprinted tissue construct that are crucial for the end application. During tissue formation these properties are influenced by the amount of cells present in the construct, their proliferation, migration and interaction with the material. A calibrated computational framework is able to predict the tissue development and maturation and to optimize the bioprinting input parameters such as the starting material, the initial cell loading and the construct geometry. In this contribution relevant bioink properties are reviewed and discussed on the example of most popular bioprinting approaches. The effect of cells on hydrogel processing and vice versa is highlighted. Furthermore, numerical approaches were reviewed and implemented for depicting the cellular mechanics within the hydrogel as well as for prediction of mechanical properties to achieve the desired hydrogel construct considering cell density, distribution and material-cell interaction.

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Element free Galerkin (EX) methods are methods for solving p a differential equations that require only nodal data and a description of the gwmeuy; no element connectivity data are needed. This makes the method very atmctive for the modeling of lhe propagation of cracks. as the number of data changes required is small and easily developed. The method is based on the use of moving least-squares interpolants with a Galerkin method, and it provides highly accurate solutions for elliptic problem. The implementation of the EFC method for problems of frachlre and static crack growth is described. Numerical examples show that accurate stress intensity factors can be obtained without any enrichment of the displacement field by a near-crack-tip singularity and that crack growth can be easily modeled since it requires hardly any remeshing.

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