Metamaterials composed of different geometrical primitives have different properties. Corresponding to the fundamental geometrical forms of line, plane, and surface, beam-, plate-, and shell-based lattice metamaterials enjoy many advantages in many aspects, respectively. To fully exploit the advantages of each structural archetype, we propose a multilayer strategy and topology optimization technique to design lattice metamaterial in this study. Under the frame of the multilayer strategy, the design space is enlarged and diversified, and the design freedom is increased. Topology optimization is applied to explore better designs in the larger and diverse design space. Beam-plate-shell-combined metamaterials automatically emerge from the optimization to achieve ultrahigh stiffness. Benefiting from high stiffness, energy absorption performances of optimized results also demonstrate substantial improvements under large geometrical deformation. The multilayer strategy and topology optimization can also bring a series of tunable dimensions for lattice design, which helps achieve desired mechanical properties, such as isotropic elasticity and functionally grading material property, and superior performances in acoustic tuning, electrostatic shielding, and fluid field tuning. We envision that a broad array of synthetic and composite metamaterials with unprecedented performance can be designed with the multilayer strategy and topology optimization.

This paper deals with topological design optimization of elastic, continuum structures without damping that are subjected to time-harmonic, design-independent external dynamic loading with prescribed excitation frequency, amplitude and spatial distribution. The admissible design domain, the boundary conditions, and the available amount(s) of material(s) for single- or bi-material structures are given. An important objective of such a design problem is often to drive the resonance frequencies of the structure as far away as possible from the given excitation frequency in order to avoid resonance and to reduce the vibration level of the structure. In the present paper, the excitation frequency is defined to be ‘low’ for positive values up to and including the fundamental resonance frequency of the structure, and to be ‘high’ for values beyond that. Our paper shows that it may be very important to consider different design paths in problems of minimum dynamic compliance in order to obtain desirable solutions for prescribed excitation frequencies, and a so-called ‘incremental frequency technique’ (IF technique) is applied for this. Subsequently, the IF technique is integrated into an extended, systematic method named as the ‘generalized incremental frequency method’ (GIF method) that is developed for gradient based dynamic compliance minimization for not only ‘low’, but also ‘high’ excitation frequencies of the structure. The GIF method performs search for and determination of the solution to the minimum dynamic compliance design problem, but this is subject to the complexity that problems with prescribed high excitation frequencies exhibit disjointed design sub-spaces. Each of these sub-spaces is associated with a local minimum value of the dynamic compliance, so in general the ‘global optimum solution’ will have to be selected as the ‘best’ solution from among a number of local candidate solutions. Illustrative examples of application of the IF technique and the GIF method are presented in the paper.