An algorithm for the topology optimization of geometrically nonlinear structuresGomes, Francisco A.M.; Senne, Thadeu A.
doi: 10.1002/nme.4686pmid: N/A
Most papers on topology optimization consider that there is a linear relation between the strains and displacements of the structure, implicitly assuming that the displacements of the structure are small. However, when the external loads applied to the structure are large, the displacements also become large, so it is necessary to suppose that there is a nonlinear relation between strains and displacements. In this case, we say that the structure is geometrically nonlinear. In practice, this means that the linear system that needs to be solved each time the objective function of the problem is evaluated is replaced by an ill‐conditioned nonlinear system of equations. Moreover, the stiffness matrix and the derivatives of the problem also become harder to compute.
Optimizing perfectly matched layers in discrete contextsModave, A.; Delhez, E.; Geuzaine, C.
doi: 10.1002/nme.4690pmid: N/A
Perfectly matched layers (PMLs) are widely used for the numerical simulation of wave‐like problems defined on large or infinite spatial domains. However, for both time‐dependent and time‐harmonic cases, their performance critically depends on the so‐called absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyze how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one‐dimensional and two‐dimensional representative time‐dependent cases. This study highlights the advantages of the so‐called shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions. Copyright © 2014 John Wiley & Sons, Ltd.
An efficient augmented finite element method for arbitrary cracking and crack interaction in solidsLiu, W.; Yang, Q.D.; Mohammadizadeh, S.; Su, X.Y.
doi: 10.1002/nme.4697pmid: N/A
This paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary, multiple intra‐elemental discontinuities in 2D solids. The augmented finite element method (A‐FEM) employs four internal nodes to account for the crack displacements due to an intra‐elemental weak or strong discontinuity, and it permits repeated elemental augmentation to include multiple interactive cracks. It thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity, and to multiple interactive cohesive cracks, all within a single bilinear element that employs standard external nodal DoFs only. A novel elemental condensation procedure has been developed to solve the internal nodal DoFs as functions of the external nodal DoFs for any irreversible, piece‐wise linear cohesive laws. It leads to a fully condensed elemental equilibrium equation with mathematical exactness, eliminating the need for nonlinear equilibrium iterations at elemental level. The new A‐FEM's high‐fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous, and heterogeneous solids have been demonstrated through several challenging numerical examples. It is shown that the proposed A‐FEM, empowered by the new elemental condensation procedure, is numerically very efficient, accurate, and robust. Copyright © 2014 John Wiley & Sons, Ltd.