Topology optimization of hyperelastic structures with anisotropic fiber reinforcement under large deformations
Introduction
Fiber-reinforced soft materials find potential applications in many emerging engineering fields, such as soft actuators [1], [2], [3], soft robotics [4], [5] and soft tissue engineering [6], [7]. As a powerful computational design tool, topology optimization offers a systematic approach to design fiber-reinforced soft materials to realize their multi-functionality. In general, topology optimization formulations involving fiber orientation design can be categorized into two types. In the first type, the fiber orientation at each point can continuously vary from (e.g., ) to (e.g., ), see, e.g., [8], [9], [10], [11], [12]. The second type aims to find the best distribution of fiber angles among a discrete set of fiber orientations defined a priori. In this work, we focus on formulations of the second type. In order to ensure that the fiber orientation converges to a discrete set of candidate angles, a fiber orientation interpolation scheme is typically needed. Several fiber interpolation schemes have been proposed in the literature. The representative schemes include discrete material optimization (DMO) [13], shape function with penalization (SFP) [14] and bi-value coding parameterization (BCP) [15], and normal distribution function optimization (NDFO) and its variants [16], [17]. Despite the above-mentioned progress, establishing a topology optimization formulation for designing fiber-reinforced soft materials and structures still needs further development. While the most studies have been limited to linear elastic behavior (for both matrix and fiber), formulations capable of accounting for nonlinear behaviors of both matrix and fiber phases under large deformations and simultaneously optimizing both phases are rarely considered (see, however, Refs. [18], [19], [20] for work on simultaneous optimization of linear matrix and fiber phases, and Ref. [21] for work on optimizing fiber orientations in (non-designable) hyperelastic matrix under large deformations).
From the optimization perspective, topology optimization considering multiple pre-defined candidate fiber orientations naturally involves multiple sets of design variables, making the design update process a computationally expensive task for general-purpose update schemes. In Refs. [22], [23], a design variable update scheme, named the Zhang–Paulino–Ramos (ZPR), was proposed considering multiple materials with both linear and nonlinear properties. The ZPR exploits the separable feature of the convex approximated subproblem at each optimization step and derives a decoupled update formula for the design variables associated with each volume constraint. The ZPR update has been mostly used in multi-material topology optimization consisting of multiple volume constraints and has not been explored in topology optimization problems with matrix phase and fiber reinforcements.
In this work, we put forth a general topology optimization framework to design hyperelastic structures with anisotropic fiber reinforcements under large deformations. The contributions of our proposed framework are as follows:
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First, the propsoed framework simultaneously optimizes both material distribution in matrix phase and orientation distribution of the underlying fiber reinforcements under large deformations. In addition, we take into account nonlinear behaviors of both matrix and fiber phases.
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Second, we introduce a design parameterization, where the matrix and fiber phases are represented by separate design variables. The advantage of this parameterization is that it allows for the seamless incorporation of geometric restrictions (e.g., symmetry constraints) into the matrix phase without influencing the fiber orientation distributions. To integrate the representations of matrix and fiber phases, we propose a general anisotropic material interpolation scheme that relates the stored-energy function of the resulting composites to the design variables associated with both phases.
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Third, inspired by the ZPR design update scheme for multi-material topology optimization [22], a decoupled design variable update scheme is proposed to enable separable and parallelizable update of the design variables associated with the density field and each fiber orientation field. We further derive explicit update formula (i.e., no inner iterations are needed) for the design variables associated with each fiber orientation field.
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Last but not the least, we adopt the virtual element method (VEM) in conjunction with a tailored adaptive refinement and coarsening scheme [24], [25] to efficiently solve the boundary value problem of finite elasticity. As an emerging computational method, the VEM has attracted growing interests in the fields of computational mechanics, see, e.g., [26], [27], [28], [29], [30], [31], [32], and topology optimization [33], [34], [35]. The VEM has a unique feature of handing arbitrary element geometries and, thus, offers an ideal platform for flexible and adaptive refinement and coarsening schemes to accelerate topology optimization without sacrificing design resolution.
The remainder of this paper is organized as follows. Section 2 briefly reviews the constitutive models for transversely isotropic materials and the variational principle of finite elasticity. Section 3 introduces the VEM formulation to simulate the nonlinear elastic response of fiber-reinforced material under large deformations, including a simple example to verify the convergence of the VEM approximation. In Section 4, we introduce the general topology optimization framework for the design of fiber-reinforced hyperelastic structures with emphasis on design parameterization, the fiber orientation and anisotropic material interpolation schemes, and the energy interpolation approach. In Section 5, a decoupled design variable update scheme is proposed and explicit update formulas are derived for the design variables associated with the fiber orientation fields. Section 6 presents design problems in conjunction with three objective functions and two fiber interpolation schemes to demonstrate the effectiveness of the proposed topology optimization framework. Section 7 provides several concluding remarks of the present work. The sensitivity analysis of the proposed topology optimization framework is provided in Appendix.
Section snippets
Finite elasticity considering transversely isotropic solids
This section briefly reviews the constitutive models of transversely isotropic materials [36] and the variational formulation adopted in this work. We adopt a Lagrangian description and neglect the presence of body force.
As illustrated in Fig. 1(a), let us consider a solid that occupies in its undeformed configuration (which is assumed to be stress-free). On the boundary, a traction field and a displacement field are applied on and , respectively, such that and .
The VEM formulation for modeling fiber-reinforced elastomers
Consider a discretization of the domain into non-overlapping elements. By convention, we denote the discretized domain as . We also use and to denote a generic edge and element, respectively, with and denoting the length and area of and , respectively. An illustration of the discretized domain , a generic element and a generic edge is provided in Fig. 1(b). We adopt the VEM to discretize and approximate the weak form (9) [42], [43]. Alternatively, one can also use the
Topology optimization formulation for anisotropic fiber-reinforced hyperelastic structures under large deformations
We propose a general topology optimization framework for designing fiber-reinforced hyperelastic structures together with the associated design parameterization, matrix and fiber interpolation schemes, and the energy interpolation. The sensitivity analysis of the proposed topology optimization framework is presented in the Appendix.
A decoupled fiber-matrix design variable update scheme
Traditionally, the design variables and , in the optimization formulation (22) need to be updated in a coupled fashion. Inspired by the ZPR scheme [22], we derive that the update of and each should in fact be performed in a decoupled manner analytically. Furthermore, we show that an explicit update formula for the design variables can be derived.
We first denote the design variables at the optimization step as and . We then adopt the explicit
Design examples
In this section, we present three design examples that demonstrate the effectiveness of the proposed framework in simultaneously optimizing both the material distribution of the matrix phase and the fiber orientations in the reinforcement phase.
We consider three objective functions to showcase the applicability of the proposed framework to a wide range of problems. The first example considers the classic end-compliance minimization problem [56] with the objective function of the following
Concluding remarks
We present a general topology optimization framework for the computational optimized design of hyperelastic structures reinforced by nonlinear and anisotropic fibers under large deformations. We introduce a design parameterization in which the matrix phase and fiber orientations are represented by two separate sets of design variables, thus, it allows for the direct incorporation of geometric (e.g., symmetry) constraints on the matrix phase without influencing the orientation distribution of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors X. S. Zhang and Z. Zhao would like to acknowledge the financial support from the University of Illinois at Urbana Champaign, United States . The information provided in this paper is the sole opinion of the authors and does not necessarily reflect the view of the sponsoring agencies.
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