Introduction
Optimal Design Methods
Shape optimization seeks to find the optimal design of a structure by controlling the shape of voids or subregions in the domain. By not assuming the number or location of structural elements, topology optimization is the most expressive, being able to search a wider spectrum of the design space.
Optimal Design for More Complex Problems
However, with additional design variables such as material orientation, we need to develop appropriate control methods. As we intend to develop methods that can be translated into applications, we must consider the possibility of fabrication.
Outline of Thesis
In Section 3.6 we consider the introduction of holes or voids in the domain, and show that the existence result remains valid for the blocking load objective. In the previous 2D cases, the gradients of the objective with the design variables for the initial uniform density designs were substantial.
Background: Minimum Compliance
Conclusion
Thus, the target blocking load is the ratio of the target work of activation to the compliance of the unstimulated structure. We also note that the patterns we get depend on the location of the load.
Optimal Design of Responsive Structures
Introduction
Next, in Section 3.5.2, we discuss the design of the same actuator setups, now optimized for blocking loading. Then, in Section 3.6.2, we show the formulation in a 3D setting considering the optimal design of the torsional actuator.
Background: Compliance Optimization
Roughly speaking, the introduction of a non-local term in the response function (the filter) provides compactness of the minimization of series of designs which, combined with the lower semi-continuity of the objective function, is sufficient to ensure the existence of "classical" prove solutions. We will borrow these ideas for the optimal design of responsive structures to formulate a well-posed problem.
Optimizing Responsive Structures
Precisely, the regulation provided by the filter provides poor continuity of compliance in both the driven and undriven conditions. We need the following lemma, which establishes the weak continuity of the solutions to the elliptic problem and whose proof is provided in Appendix 3.A.
Objective Functions
Blocking load is the magnitude of applied load that cancels activation. Thus, the blocking load objective leads to a structure that balances actuation work and structure stiffness.
Examples of Optimal Responsive Structures
The value of the objective increases with the allowable volume fraction of responsive material, but saturates when the volume fraction does. The designs remain broadly similar because we change the ratio between the stiffness of the responsive material and that of the structural material (Er/Es).
Optimizing Responsive Structures with Voids
Optimizing the blocking load under this loading is analogous to maximizing the blocking torque of the actuator. We consider the formulation of the previous section for two materials with voids, using identical numerical schemes. As one would expect, we see that the responsive material is arranged spirally toward the outer edges of the domain.
However, for the 3D torque actuator, the gradients of the blocking load target with the design variable φ were close to zero for a uniform density design, especially in the case of Er/Es = 1.
Conclusions
This can occur either at the boundary of the design domain or in the interior near holes or material interfaces. The second term penalizes director fields perpendicular to the boundary of the responsive structure inside the domain. Details of the numerical methods for solving the forward and adjoint problem are discussed in the following section.
For simplicity, we assume that the rest of the material properties are identical (density, hardening parameters, damage length scale).
Optimal Design of 3D Printed Soft Responsive Actuators . 60
Design for Small Strains
In addition, we will show that for a given class of objective functions, the optimal transformation strains will have the form (4.2). We also relax the transformation stresses and instead consider the convexification of the original space of transverse isotropic transformation. So far we have shown that the optimization problem is well positioned within the relaxed space of transformation voltages.
We will now show that there exist optimal solutions that have transformation strains of the form (4.2) for a certain class of objective functions.
Design for Finite Strains
The first is the sequential lower semicontinuity of the energy function, and the second is its Γ-convergence. In addition, the linearity of C and the convergence of {uk} give sequential weak lower semi-continuity of the energy function E. From ¯ the weak convergence of φk, the properties of the filter, and the strong convergence of ak in L∞ the integrand converges from W(φk, ak,u, S) pointwise to the integrand of W( ¯φ,¯a,u, S).
Having established the lower semi-continuity of conformity, we are now ready to prove the existence of optimal designs for (4,84) under a suitable assumption.
Examples of Lifting Actuators
The existence proof for two-material design in Theorem 4.3.5 can simply be extended to include voids. We consider the optimal design of a lift actuator on the rectangular 2D domain shown in Figure 4.1, with a vertical load applied at the lower right corner. We see that in the case of softer responsive material the structure contains large areas of responsive material.
Here, the finite deformation kinematics cause the elongation of these elements to cause significant vertical displacement.
Simultaneous Design with Printing Constraints
We introduce a scalar-valued function Ψ : Ω 7→ R, whose contours describe the print path in the domain of the response structure. While the free design acts as a push actuator with the responsive material distributed along the bottom edge with near-vertical director alignment, the print-aware design acts primarily as a pull actuator with the responsive material distributed along the top of the structure. In addition, the nematic routers in the print-aware package are aligned with the main support of the active material.
We see that the print lines of the responsive structure follow the desired paths quite well.
Conclusion
This gives the total target angle variation. where the adjoint variables suffice for dynamic evolution 0 =. 5.15). We first consider the optimal design of a multi-material structure subjected to a relatively high impact velocity of 0.110 cL, where cL is the longitudinal wave velocity of the strong material. Target values and volume fraction of strong material are shown for each design.
An overview of the adjoint-state method for calculating the gradient of a feature with geophysical applications.
Optimal Structures for Failure Resistance Under Impact
Introduction
Then, gradient-based optimization methods are used to iteratively update the design, where sensitivities are usually calculated through the adjoint method. By assuming that the material parameters depend on a continuous design variable, we derive sensitivities through the adjoint method. We use an explicit update scheme for the adjoint displacement variable and another augmented Lagrangian method for the adjoint damage variable.
We use an adjoint method where sensitivities and adjoint relations are derived for a general objective.
Theoretical Formulation
The stationarity of this action integral gives the dynamical evolution and the kinetics of the internal variables [31]. 5.10e). U is the space of allowable displacement variations. a) is the second-order dynamical evolution of the displacement field. In addition, the convexity of the adjoint problem with respect to the entire variable set {ξ, γ, µ, b} has not been established, which may lead to an ill-posed problem.
In addition, issues may arise from the possible temporal discontinuities of the damage field discussed previously in section 5.2.1.
Numerics
We now turn to the details of the numerical evolution of the adjoint problem, which must be solved backward in time using the solution to the forward problem. For efficiency, we use another extended Lagrangian formulation for the additional damage variable update. Similar to the forward problem, we use an explicit central difference scheme for the adjoint displacement variable.
Then, we implicitly update the adjacent damage variables through an alternating direction method with multipliers.
Material Interpolation
This mitigates spurious dynamic modes that may arise from artificial acoustic properties of voids [7]. Unfavorable intermediate densities: The intervention of the plastic potential must ensure that the relative yield stress is not too high in regions of intermediate density, so that the optimal solutions are dominated by completely solid or void regions. Life (ηmin), (5.58) ensuring that the relative stability of the voids is much greater than that of the solid.
Thus, the yield strain for the void regions is approximately twice that of the solid.
Examples
This is an acceptable compromise, because we are mainly interested in the spall phenomena that occur near the center of the structure, but also hinged at the boundaries. In addition, at the two largest speeds, strong material is used on the top surface under the sides of the flyer. The material parameters we use are identical to those of the previous study, with the following exceptions.
Moving from the right to the left column increases the yield strength of the tough material by 50% compared to the previous study, while moving from the top row to the bottom row increases the toughness of the tough material by 50%.
Discussion and Conclusion
The financial support of the US National Science Foundation through "Collaborative Research: Optimal Design of Responsive Materials and Structures". In this thesis, we investigated optimal design for applications exploiting recent advances in materials and fabrication. Through these studies, we have addressed major challenges related to the broader goal of optimal design for emerging materials and manufacturing technologies.
Another pressing engineering problem that we can explore through optimal design is that of energy conversion or storage devices.
Conclusion
Extensions
We can extend the work on soft responsive actuators to additionally consider the response time in the goal. After developing high-fidelity models to capture these multi-physics interactions in a use case environment, we can apply optimal design in this environment. We can also explore a similar problem of energy conversion devices, where a reaction-diffusion process converts inert gas to usable fuel.
Here we can use PDE-based optimization methods to construct a method corresponding to optimal design, where the design parameters are now the weights and biases of a neural-net constitutive law.
Challenges Moving Forward
These systems are complicated by physical mechanisms such as ion transport along with solid deformation and material failure. Here, we could explore the design of energy storage devices such as solid state batteries with greatly improved performance and lifetime. Finally, the tools developed for optimal design can be applied to other PDE-based optimization problems.
We have recently begun to explore this through a mechanistically consistent formulation that explains history dependency.
