Anisotropic elastic metamaterials
Elastic metamaterial designs are provided, such as an acoustic radiator or sound partition, with non-spherical shapes or apertures defined in unit cells of an elastic medium. A method for making the same includes determining a set of boundary conditions for a plurality of non-spherical shapes/apertures defined in the elastic medium, and using a gradient-based algorithm to optimize a porous media model domain for the elastic medium, where porosity is related to size dimensions of the non-spherical shape/aperture and an anisotropic elastic modulus is related to an angle of orientation of the non-spherical shape/aperture. The method may include optimizing an objective function, and obtaining a grayscale design that relates to the porosity and the anisotropic elastic modulus. Reaction diffusion equations may be used with the grayscale design to obtain a pattern for the non-spherical shapes/apertures. Methods of manufacturing may include multi-material additive manufacturing techniques.
1 . A method for manufacturing an elastic metamaterial for an acoustic radiator, the method comprising:
defining an array of unit cells that form the acoustic radiator, the array including an elastic medium comprising a solid phase material;
determining a set of boundary conditions for a plurality of non-spherical apertures defined in the elastic medium of the array, with each non-spherical aperture disposed within a boundary defined by a single respective unit cell of the array;
using a gradient-based algorithm to optimize a porous media model domain for the elastic medium, where porosity is related to size dimensions of the plurality of non-spherical apertures and an anisotropic elastic modulus is related to an angle of orientation of the non-spherical aperture;
optimizing an objective function, and obtaining a grayscale design that relates to the porosity and the anisotropic elastic modulus;
using reaction diffusion equations to de-homogenize the grayscale design to obtain a pattern for the non-spherical apertures; and
manufacturing the elastic metamaterial by forming the pattern for the non-spherical apertures in the solid phase material such that the non-spherical apertures extend through the solid phase material.
2 . The method according to claim 1 , wherein the gradient-based algorithm comprises a topology optimization problem solved according to constitutive laws associated with a linearly elastic medium.
3 . The method according to claim 2 , wherein the topology optimization problem is solved maximizing or minimizing a spectral displacement variable or set of variables of the elastic medium.
4 . The method according to claim 3 , wherein the spectral displacement variable is proportional to a structure root mean square velocity at one or both of a predetermined point and predetermined frequency.
5 . The method according to claim 2 , wherein the gradient-based algorithm comprises at least one of a method of moving asymptotes (MMA) optimizer for the topology optimization problem and a globally convergent method of moving asymptotes (GCMMA) optimizer for the topology optimization problem.
6 . The method according to claim 1 , comprising using an anisotropic diffusion tensor with two-component reaction diffusion equations.
7 . The method according to claim 6 , wherein the step of using reaction diffusion equations with the grayscale design to obtain a pattern of non-spherical apertures comprises extracting a unit cell porosity magnitude plus a tensor-expression of the anisotropic elastic modulus.
8 . The method according to claim 6 , comprising repeatedly solving the reaction diffusion equations for a time period and alternatively using weakly anisotropic and strongly anisotropic diffusion tensors.
9 . The method according to claim 1 , wherein a frequency response of the acoustic radiator is variable based on a single-material or multi-material selection design of the acoustic radiator.
10 . The method according to claim 1 , wherein the step of determining a set of boundary conditions for the plurality of non-spherical apertures comprises using at least one look-up table mapping grayscale design information and unit cell designs to a size and orientation of the non-spherical apertures.
11 . The method according to claim 10 , wherein the x-axis of the mapping corresponds to a width dimension of the non-spherical aperture in the unit cell, the y-axis corresponds to a height dimension of the non-spherical aperture, and the z-axis is an elastic modulus tensor component.
12 . The method according to claim 11 , wherein the at least one look-up table is based on data obtained from varying the width dimension and the height dimension of the non-spherical aperture in the unit cell over a range of values and calculating the tensor component.
13 . A method for manufacturing an elastic metamaterial for an acoustic radiator, the method comprising:
defining an array of unit cells that form the acoustic radiator, the array including an elastic medium comprising a solid phase material selected from the group consisting of polymers, polymer-based materials, metals, and composite materials, and having an x-axis defining a longitudinal direction, a y-axis defining a transverse direction with respect to the x-axis, and a z-axis perpendicular to both the x-axis and the y-axis;
determining a set of boundary conditions for a plurality of non-spherical apertures defined in the elastic medium of the array, with each non-spherical aperture disposed within a boundary defined by a single respective unit cell of the array;
using a gradient-based algorithm to optimize a porous media model domain for the elastic medium, the gradient-based algorithm comprising a topology optimization problem solved according to constitutive laws associated with a linearly elastic medium, and where porosity is related to size dimensions of the non-spherical aperture and an anisotropic elastic modulus is related to an angle of orientation of the non-spherical aperture;
optimizing an objective function, and obtaining a grayscale design that relates to the porosity and the anisotropic elastic modulus;
using reaction diffusion equations to de-homogenize the grayscale design to obtain a pattern for the non-spherical apertures; and
manufacturing the elastic metamaterial by forming the pattern for the non-spherical apertures in the solid phase material such that the non-spherical apertures extend through the solid phase material.
14 . The method according to claim 13 , wherein the topology optimization problem is solved maximizing or minimizing a spectral displacement variable or set of variables of the elastic medium.
15 . The method according to claim 14 , wherein the spectral displacement variable is proportional to a structure root mean square velocity at one or both of a predetermined point and predetermined frequency.
16 . The method according to claim 13 , wherein the gradient-based algorithm comprises at least one of a method of moving asymptotes (MMA) optimizer for the topology optimization problem and a globally convergent method of moving asymptotes (GCMMA) optimizer for the topology optimization problem.
17 . The method according to claim 13 further comprising using an anisotropic diffusion tensor with two-component reaction diffusion equations.
18 . The method according to claim 17 , wherein the step of using reaction diffusion equations with the grayscale design to obtain a pattern of non-spherical apertures comprises extracting a unit cell porosity magnitude plus a tensor-expression of the anisotropic elastic modulus.
19 . The method according to claim 17 further comprising repeatedly solving the reaction diffusion equations for a time period and alternatively using weakly anisotropic and strongly anisotropic diffusion tensors.
20 . The method according to claim 13 , wherein a frequency response of the acoustic radiator is variable based on a single-material or multi-material selection design of the acoustic radiator.