The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Advances in 4D Printing: Materials and Applications

Manufacturing-oriented topology optimization has been extensively studied the past two decades, in particular for the conventional manufacturing methods, for example, machining and injection molding or casting. Both design and manufacturing engineers have benefited from these efforts because of the close-to-optimal and friendly-to-manufacture design solutions. Recently, additive manufacturing (AM) has received significant attention from both academia and industry. AM is characterized by producing geometrically complex components layer-by-layer, and greatly reduces the geometric complexity restrictions imposed on topology optimization by conventional manufacturing. In other words, AM can make near-full use of the freeform structural evolution of topology optimization. Even so, new rules and restrictions emerge due to the diverse and intricate AM processes, which should be carefully addressed when developing the AM-specific topology optimization algorithms. Therefore, the motivation of this perspective paper is to summarize the state-of-art topology optimization methods for a variety of AM topics. At the same time, this paper also expresses the authors' perspectives on the challenges and opportunities in these topics. The hope is to inspire both researchers and engineers to meet these challenges with innovative solutions.

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Current and future trends in topology optimization for additive manufacturing

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Nonlinear Finite Elements For Continua And Structures

We describe an approach to print composite polymers in high-resolution three-dimensional (3D) architectures that can be rapidly transformed to a new permanent configuration directly by heating. The permanent shape of a component results from the programmed time evolution of the printed shape upon heating via the design of the architecture and process parameters of a composite consisting of a glassy shape memory polymer and an elastomer that is programmed with a built-in compressive strain during photopolymerization. Upon heating, the shape memory polymer softens, releases the constraint on the strained elastomer, and allows the object to transform into a new permanent shape, which can then be reprogrammed into multiple subsequent shapes. Our key advance, the markedly simplified creation of high-resolution complex 3D reprogrammable structures, promises to enable myriad applications across domains, including medical technology, aerospace, and consumer products, and even suggests a new paradigm in product design, where components are simultaneously designed to inhabit multiple configurations during service.

Direct 4D printing via active composite materials

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.

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Level Set Topology Optimization of Printed Active Composites

A new variational method for selective mass scaling is proposed. It is based on a new penalized Hamilton's principle where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices. In this framework new mass matrices with desired properties can be constructed. It is demonstrated how usage of these non-diagonal mass matrices decreases the maximum frequency of the discretized system and allows for larger steps in explicit time integration. At the same time the lowest eigenfrequencies in the range of interest and global structural response are not significantly changed. Results of numerical experiments for two-dimensional and three-dimensional problems are discussed.

Variational methods for selective mass scaling

Summary
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. Copyright © 2014 John Wiley & Sons, Ltd.

Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling

The problem of optimal selective mass scaling for linearized elasto-dynamics is discussed. Optimal selective mass scaling should provide solutions for dynamical problems that are close to the ones obtained with a lumped mass matrix, but at much smaller computational costs. It should be equally applicable to all structurally relevant load cases. The three main optimality criteria, namely eigenmode preservation, small number of non-zero entries and good conditioning of the mass matrix are explicitly formulated in the article. An example of optimal mass scaling which relies on redistribution of mass on a global system level is constructed. Alternative local mass scaling strategies are proposed and compared with existing methods using one modal and two transient numerical examples.

Local and global strategies for optimal selective mass scaling

SUMMARY An alternative spatial semi-discretization of dynamic contact based on a modified Hamilton’s principle is proposed. The modified Hamilton’s principle uses displac ement, velocity and momentum as variables, which allows their independent spatial discretization. Al ong with a local static condensation for velocity and momentum it leads to an approach with a hybrid-mixed consistent mass matrix (HMCMM). An attractive feature of such a formulation is the possibility to construc t hybrid singular mass matrices (HSMM) with zero components at those nodes where contact is collocated. This improves numerical stability of the semidiscrete problem: the differential index of the underlying differential-algebraic system is reduced from 3 to 1 and spurious oscillations in the contact pressure, whi ch are commonly reported for formulations with Lagrange multipliers, are significantly reduced. Resu lts of numerical experiments for truss and Timoshenko beam elements are discussed. In addition, the properties of the novel discretization scheme for an unconstrained dynamic problem are assessed by a dispersion analysis. Copyright c 2010 John Wiley & Sons, Ltd. Received . . .

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Anton Tkachuk

Papers

Level Set Topology Optimization of Printed Active Composites

Variational methods for selective mass scaling

Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling

Local and global strategies for optimal selective mass scaling

Hybrid-mixed discretization of elasto-dynamic contact problems using consistent singular mass matrices