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Ceramic materials show good mechanical properties. However, the brittleness makes it difficult to predict their dynamic behavior of ceramic materials, and further the discrete nature of brittle failure requires a three-dimensional computation. In this paper, a computational model was developed for analyzing high velocity impact situations of brittle materials, incorporating with node separation scheme to describe the distinctive behavior of a brittle failure. This material behavior model was applied in a three-dimensional explicit code with Lagrangian description, which was developed to compute problems of dynamic brittle failure using four-node tetrahedral elements. To effectively handle the complex contact situations, involving multiple collisions and self-contact of severely deformed and fragmented bodies, careful considerations were paid to the global and local contact searching. In addition, an edge contact algorithm was developed to handle contacts between sharp edges. To validate the developed computational model and three-dimensional explicit code, oblique penetration into sandwiched ceramic plate by long rod was simulated, and the results were compared with experiments.

We present a mixed adaptive variational multiscale method for solving elliptic second-order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Målqvist,^15–17 to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates both for linear functionals and the energy norm. Based on the estimates we propose adaptive algorithms for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two-dimensional slice of an oil reservoir.

We propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyze general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalized solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration, we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behavior for heat-exchangers. We also provide general expression for the infinite asymptotic behavior of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analyzed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes.

Hyperelastic models are of particular interest in modeling biomaterials. In order to implement them, one must derive the stress and elasticity tensors from the given potential energy function explicitly. However, it is often cumbersome to do so because researchers in biomechanics may not be well-exposed to systematic approaches to derive the stress and elasticity tensors as it is vaguely addressed in literature. To resolve this, we present a framework of a general approach to derive the stress and elasticity tensors for hyperelastic models. Throughout the derivation we carefully elaborate the differences between formulas used in the displacement-based formulation and the displacement/pressure mixed formulation. Three hyperelastic models, Mooney–Rivlin, Yeoh and Holzapfel–Gasser–Ogden models that span from first-order to higher order and from isotropic to anisotropic materials, are served as examples. These detailed derivations are validated with numerical experiments that demonstrate excellent agreements with analytical and other computational solutions. Following this framework, one could implement with ease any hyperelastic model as user-defined functions in software packages or develop as an original source code from scratch.

This paper presents the theory and applications of the Hellinger-Reissner mixed formulation for the nonlinear frame element with lateral deformable supports. The governing differential equations of the problem (strong form) are derived first. Then, the Hellinger-Reissner mixed frame element (weak form) is formulated to solve for the numerical solution of the problem. Tonti' s diagrams are employed to conveniently represent the equations governing both strong and weak forms of the problem. Finally, a numerical example is used to show that the Hellinger-Reissner mixed element is much more accurate than the classical displacement-based element. The nonlinear frame model proposed in this paper has practical applications in modelling the soil-pile structural system, geosynthetics/fiber-glass reinforcement of foundation soils, beam on deformable foundations, etc.

For the obstacle problem: