Design of Viscoelastic Damped Plate Structures with a Multi-Material Topology Optimization Method

A viscoelastic damping layer attached to the plate structure can control vibrations but increases mass. Involving multiple viscoelastic materials, the contribution of different materials needs to be evaluated and allocated. In the paper, a topology optimization model is proposed to find distributions of different viscoelastic materials adopting a multi-material level set method, for maximizing single-order or multi-order modal loss factors. Limited by material volume fraction, the variational principle is applied to establish the velocity field, and the problem is solved by a gradient-based method. Numerical examples are performed to illustrate that loss factors of composite plate structures can be maximized with optimal distributions obtained from the multi-material level set method.


Introduction
The vibration level of the structure is one of the quality evaluation criteria of the product, which penetrates into every process from design to production.One of the techniques to control the vibration of composite plate structures that are widely used in mechanical systems is to perform surface damping treatment.
In order to control the system mass, Ansari et al. [1] took the classical level set topology optimization to determine the distribution for constrained damping patches attached to the board, aiming at maximizing loss factors.And later, Kolk et al. [2] simultaneously optimized the distribution scheme of the substrate and the viscoelastic material layer together, for maximum modal loss factors at target frequencies.
Among structural topology optimization (STO) methods, the level set method (LSM) could be a superior for description of boundaries [3,4].Farther, parametric methods are proposed to alleviate the difficulty of solving level-set equations [5][6][7].The discrete adjoint method is used to deal with sensitive boundaries to gain sensitivity with greater precision [8].As application and problems become more complex, level set description models involving multiple materials have also been proposed, adapting the LSM to more detailed situations [9,10].
In the paper, a STO model with multi-material level set (MM-LS) method has been proposed, simplified by a parameterization method, to optimize the distribution of viscoelastic materials on a composite plate.Finally, numerical cases are implemented to illustrate that the model is effective and feasible.

Vibration Equation of the Plate
In the studied problem, a rectangular composite plate, as shown in figure 1, whose substrate is made of metal material, is covered with a free layer damping made of viscoelastic material.
The complex constant modulus model is taken to describe the viscoelastic material constitutive relation of the free damping layer.Damping characteristics of viscoelastic materials are expressed in terms of complex stiffness.Under the action of external force, dynamic equation of the composite plate structure is ( ) where M means the mass matrix, R K and I K are the real part and the imaginary part of the stiffness matrix, respectively.F means external loads, x shows the displacements.The characteristic equation could be where  represents the eigenvalue,  denotes the eigenvector; rr  = is the modal frequency for the rth order, r  is modal vectors for the rth order.The modal loss factor reflects the capability to dissipate energy at a certain modal frequency and is an important indicator of the structural modal characteristics of the system.The modal loss factor at rth order will be shown:

Optimization Model with Multi-Material Level Set Method (MM-LS)
The LSM is expert in describing dynamic changes of structure boundaries and is one of the best interface tracking techniques.The level set function can be expressed as: where ( ) When multiple materials are employed, the MM-LS method provides a model that describes m kinds of materials and a void field with m level set functions, as shown in figure 2, drawing on ideas of multi-material processing by the SIMP.The MM-LS model is represented as follows: According to this, a STO model would be written as: ( ) .. 0 ( ) , ( 1, 2,..., ) When solved, i  could be replaced with a parameterized interpolation based on GSRBF.N interpolation points and a set of multiple quadratics (MQ) splines interpolation basis functions are selected to fit the level set function, then level set functions can be parameterized and expressed: where () i g x and () i t  represent the interpolation basis function and the interpolation coefficient at the ith node, respectively.Then the H-J equation becomes ( ) The original level set equation is converted from partial differential equation to ordinary differential equation, which is the advantage of parameterized level set method and can effectively alleviate the difficulty of numerical calculation.

Sensitivity Analysis
Introducing the Lagrangian multiplier, the augmented Lagrangian equation of the original problem with volume constraints is 1 ( , ) ( ) According to the variational principle, the variation of the augmented Lagrangian equation is where k  represents the kth level set function, k H represents the Heaviside function of k  , k l  is the slight perturbation of i  at the normal direction.Then velocity fields of level set functions are calculated by Euler-Lagrange equations of Equation ( 10).Because the level set equation has been expressed as an ordinary differential equation, the first-order advance Euler method or other gradient methods can be used for numerical calculation.Numerical examples in the next section will be performed with the purpose of determining the distribution with two viscoelastic materials on the plate.

Optimization Steps
Step 1: establish system equilibrium equations and perform finite element discretization; Step 2: initialize level set function 1 and level set function 2, determine optimization parameters, then calculate objective function and constraints; Step 3: sensitivity analysis by augmented Lagrangian method and variational principle to obtain velocity fields; Step 4: update level set functions for the next step by step size and obtain the objective value and volume fractions.
Step 5: whether the maximum difference between the targets of 5 adjacent iterations is less than the allowable residual, and whether the current volume fraction satisfies the constraint?If all are satisfied, the optimization is terminated.If not, return to Step 3.

Plate Structure with Free Damping
Numerical examples are performed to illustrate the MM-LS model for topology optimization.A rectangular plate with fixed constraints on all four sides is considered, which is covered with a free damping layer of mixed distribution of two viscoelastic materials.In the limit of the amount of the two viscoelastic materials, we expect to find a reasonable distribution area of the two materials to maximize one-order or multi-orders modal loss factors for the whole structure, as shown in figure 1.
In examples, the length of the plate is 0.4 m, the width is 0.2 m and the thickness is 0.001 m.The thickness of the damping layer is 0.002 m.Other materials' properties are present in table 1.We restrict volume fractions of two materials as 30% respectively.Finite element method will be adopted, and there are 40*20 quadrilateral elements.The interpolation points of level set functions are made to coincide with the grid nodes for FEM to facilitate the calculation.

Results and Discussions
First, we focus on the modal loss factor of a single order, such as the first order.The material distribution changes during the iteration process are shown in figure 3. The two colors represent the two viscoelastic materials respectively, and the blank represents only the substrate.The changes in objective and volume restrictions during iterations could be revealed in figure 4.  It's visible in figure 3 that during optimization process, material 2 (yellow area) with stronger energy dissipation capacity is gradually concentrated where the amplitude is large according to the first mode shape of the plate.Material 1 (blue area) is distributed around the periphery of material 2, which is also an area with large strain energy.Such a distribution scheme can maximize the energy dissipation capability of the material, as shown in figure 4(a).In the iterative process, in initial iterative steps, the viscoelastic material is rapidly concentrated in the region with large strain energy, so that the modal loss factor increases rapidly.Then, limited by volume restrictions, material with relatively small utility in this region begins to be reduced, the objective function decreases slightly.However, compared to the initial distribution, the final result undoubtedly achieves a larger target energy consumption with a smaller amount of material, indicating that the optimization is effective.
Subsequently, we consider both the first-order and fourth-order modal loss factors and assign a weight of 1/2 to each.In figure 5, the optimized material layout combines the first-order and fourthorder strain energy distribution characteristics.Similarly, the distribution characteristics of the two materials are similar to those of single-objective optimization, that is, material 1 surrounds material 2. figure 6 shows two level set functions of the two materials, respectively.In figure 7, during the initial iterative process, viscoelastic materials still move from the "weak region" to the "strong region" first, and then with the reduction of material, the objective decreases, but it is still better than the initial scheme.

Conclusion
In the paper, material distributions of the viscoelastic layer of damped plate structures are optimized with a MM-LS method for STO model to make target loss factors maximized.The parametric method is taken to obtain the global velocity field of level set functions and solve the optimization model.Numerical cases are performed, aiming at maximize single-order and multi-order modal loss factors, respectively, to give optimal distributions for two viscoelastic materials under volume fraction constraints.The results show that loss factors of damped plate structures can be maximized when viscoelastic materials are distributed as optimal schemes obtained by the proposed MM-LS optimization model.

Figure 1 .
Figure 1.Plate structure with free layer damping.

Figure 2 .
Figure 2. Description of level set functions for multiple materials.

H
 is the Heaviside function of the ith level set function i  , i D is the material property of the ith material, in this paper, M or K .

Figure 3 .
Figure 3. Evolution of the free layer damping.

Figure 4 .
Figure 4. Modal loss factor and volume restrictions during iterations.