Multimaterial polymer printers allow the placement of different material phases within a composite, where some or all of the materials may exhibit an active response. Utilizing the shape memory (SM) behavior of at least one of the material phases, active composites can be three-dimensional (3D) printed such that they deform from an initially flat plate into a curved structure. This paper introduces a topology optimization approach for finding the spatial arrangement of shape memory polymers (SMPs) within a passive matrix such that the composite assumes a target shape. The optimization approach combines a level set method (LSM) for describing the material layout and a generalized formulation of the extended finite-element method (XFEM) for predicting the response of the printed active composite (PAC). This combination of methods yields optimization results that can be directly printed without the need for additional postprocessing steps. Two multiphysics PAC models are introduced to describe the response of the composite. The models differ in the level of accuracy in approximating the residual strains generated by a thermomechanical programing process. Comparing XFEM predictions of the two PAC models against experimental results suggests that the models are sufficiently accurate for design purposes. The proposed optimization method is studied with examples where the target shapes correspond to a plate-bending type deformation and to a localized deformation. The optimized designs are 3D printed and the XFEM predictions are compared against experimental measurements. The design studies demonstrate the ability of the proposed optimization method to yield a crisp and highly resolved description of the optimized material layout that can be realized by 3D printing. As the complexity of the target shape increases, the optimal spatial arrangement of the material phases becomes less intuitive, highlighting the advantages of the proposed optimization method.
Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non-diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element-wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed.A. TKACHUK AND M. BISCHOFF Several algebraic methods for SMS are proposed in the literature. Mass penalty in thickness direction of solid-shell finite elements is proposed in [2,3]. This method adds artificial mass to the LMM for a relative motion in thickness direction for the nodes in the stacks of solid-shells. Such algebraic blocks reduce the highest thickness modes of the structure and at the same time are orthogonal to the bending modes, which allows accurate solutions for thin-walled structures. Stiffness proportional mass scaling ( ı D˛K) is described in [1,[4][5][6]. This method is accurate and allows preservation of the eigenvectors of the structure, but in case of non-linear problems, it requires reassembly of the scaled mass matrix for several times during the computation, because the stiffness matrix changes under large rotations and deformations [5]. The overhead introduced by reassembly is the main disadvantage of stiffness proportional mass scaling and the reason for not implementing the method in commertial finite element (FE) codes. This problem is partially resolved by the method described by Olovsson in [1]. This method uses an algebraically built template for added mass ı that does not change upon deformation and rotations. The kernel of this added mass ı includes the translational rigid body modes of individual elements, which mimics the stiffness proportional mass scaling. As a result of such a construction, the translational mass of individual elements is preserved. In addition, the method is easily implementable, and it is available in commercial codes, for example, LS-DYNA and RADIOSS. The implementation details are given in [7,8]. Further algebraic SMS methods scale the inertia of incom...
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