## 1 Introduction  
The generalized [PERSON]'s elasticity law is not valid if a structure undergoes large deformation, for which the force-displacement (F-D) curve exhibits nonlinear behavior due to geometric nonlinearity [1; 2; 3]. Geometric nonlinearities can be the softening type, where variations of displacement increase for constant increments of the external force, or hardening types, where the variations decrease. For instance, a thin plate is generally governed by hardening nonlinearity [3]. [PERSON] and [PERSON] [4] developed the topology optimization method, which has been applied to a wide range of areas over the last two decades, from mechanics to fluid dynamics, acoustics, and electrical dynamics. Most of these studies have been conducted using linear finite element analysis. However, [PERSON] et al. [5] dealt with an optimization problem concerning geometric nonlinearity due to large deformation, and many studies have since been introduced for topology optimization problems of mechanical structures associated with geometrically nonlinear modeling [6; 7; 8]. [PERSON] and [PERSON] [9] conducted one of the first studies of an optimal design problem that traces a prescribed force-displacement curve. They introduced one-step and multi-step approaches separately and they compared the results. [PERSON] et al. [10] and [PERSON] and [PERSON] [11] conducted studies to trace a prescribed curve involving snap-through behavior. They used prescribed values on the curve as a constraint in the optimal design problem. In addition, to take into account snap-through and snap-back behaviors on the force-displacement curve, they built their method on a modified arc-length procedure instead of the Newton-Raphson procedure, which is normally used in nonlinear analysis. [PERSON] et al. [12] developed a method to find an optimal layout tracing a prescribed curve defined on the reaction force and displacement plane through topology optimization. Recently, [PERSON] et al. [13] performed a study on stiffness optimization considering the secant matrix and tangent matrix, which can be defined from the F-D curve. The use of a small number of target points can be expected to decrease the computational cost. However, if the nonlinearity in the prescribed F-D curve differs from the inherent nonlinearity in the initial design, for example, the hardening nonlinearity for a thin plate, it can be impossible to find a layout that thoroughly follows the prescribed curve for the softening nonlinearity in the optimization problem with a small number of target points. For such cases, the slope constraint not only improves the convergence in the optimization design process but also decreases the computation cost in the Newton-Raphson stage. The details are discussed in the case study section. Moreover, in topology optimization studies dealing with large deformation, low stiffness elements can undergo extensive distortion during the optimal design process, which leads to numerical instabilities in the nonlinear analysis when using the Newton-Raphson method [14; 15; 16]. To circumvent the numerical instability induced by low stiffness elements, [PERSON] et al. [5] and [PERSON] et al. [17] excluded internal nodal forces that were surrounded by low stiffness elements during the optimization process. [PERSON] and [PERSON] [18] parameterized the connectivity by considering every element to be connected by imaginary springs. Recently, [PERSON] et al. [19] proposed an energy interpolation scheme using a threshold projection function. In the present work, we propose a methodology to compute the layout of a structure that traces a predefined F-D curve in which softening, hardening, or both nonlinearities are involved through density-based topology optimization. We employ the energy interpolation scheme to mitigate the numerical instability. In addition, filtering is used to ensure a global convergent design in the topology optimization, through sensitivity filtering [20], density filtering [15], and the binary bat algorithmic filter [21] approaches.