Chapter IV: Optimal Design of 3D Printed Soft Responsive Actuators . 60
4.4 Examples of Lifting Actuators
a,
φ∈Dφ, ρ∈Dinfρ, a∈Da O(φ, ρ, a) := C(φ, ρ, a,1)
C(φ, ρ, a, r) +P(a), (4.96) where we re-use the definitions of the design spaces from (4.64) and (4.86).
The existence proof for two-material designs in Theorem 4.3.5 may be simply extended for the inclusion of voids.
Theorem 4.3.6. SupposeWp is isotropic and polyconvex withWp(B)≥C1kBkpF− C2, for p > 3, Wl is of the form (4.67) with W˜l as the Mooney-Rivlin energy density from (4.68), and t∈L1(∂tΩ). Consider
O(φ, ρ, a) := C(φ, ρ, a,1)
C(φ, ρ, a, r)+P(a). (4.97) If the prescribed boundary displacementsu0 = 0and there exists some{φ,˜ ρ,˜ ˜a} ∈ Dφ× Dρ× Da such that O( ˜φ,ρ,˜ ˜a) ≤ 0, then there exists an optimal design {φ,¯ ρ,¯ ¯a} ∈ Dφ× Dρ× Da such that
O( ¯φ,¯a) = inf
φ∈Dφ, a∈Da
O(φ, a). (4.98) Proof. The proof follows almost exactly as Theorem 4.3.5, where the associated lemmas and theorem may be simply extended to the case of the additional design variable.
Figure 4.1: Rectangular domain of length L and height h for the 2D lifting actuator. Here, we apply a downward force t to the bottom right corner.
4.4.1 2D Lifting Actuator
Small Strains To simplify the computations in the small strain setting, we parameterize the director field through a single scalar angle θ : Ω 7→ [0, π) such that
a(x) = cos(θ(x))e1+ sin(θ(x))e2, (4.99) where {e1, e2} is the standard basis in R2. Then, for optimizing a structure with voids, we consider
φ∈Dφ, ρ∈Dinfρ, θ∈[0,π) O(φ, ρ, a(θ)) := ¯O(C(φ, ρ, a, S1),C(φ, ρ, a, S2)). (4.100) We consider the optimal design of a lifting actuator on the 2D rectangular domain shown in Figure 4.1, with a vertical load applied to the bottom right- hand corner. We look to minimize the compliance ratio
φ∈Dφ, ρ∈Dinfρ, θ∈[0,π) O(φ, ρ, a(θ)) = C(φ, ρ, a,1)
C(φ, ρ, a,0). (4.101) We discretize the domain with a 80×40 mesh and consider a standard Galerkin finite element formulation with first order elements [12]. We set the densities φ and ρ to be constant on each element, as well as the material orientation angle θ. After computing the actuated and un-actuated displacement for a given design and calculating the objective, sensitivities are computed using the adjoint method. Then, we update the design through the gradient-based Method of Moving Asymptotes (MMA) [9]. This is implemented in the C++
deal.ii finite element library [6].
Figure 4.2 shows the converged designs for varying elastic modulus ratios for the responsive vs passive material, Er/Es. We use a fixed Poisson ratio of
Figure 4.2: Converged designs for the rectangular lifting actuator of Figure 4.1 in the small strain setting for varying stiffness ratios for the active vs passive material. The red indicates responsive material while blue denotes passive.
The director fielda is shown in black lines.
ν = 0.48, and aspect ratio L/h = 2. We see that for the case of softer re- sponsive material, the structure contains large regions of responsive material.
Conversely, for the case of stiffer responsive material, it is distributed thinly as a frame. When the stiffness of the responsive and passive material is iden- tical, the design resembles that of a standard minimum compliance structure.
Throughout all of these, the director field is oriented nearly vertical on the bottom of the structure, and horizontal at the top. This allows it to lengthen horizontally along the bottom edge and contract at the top, creating a hinging effect, and thus lifting the load.
Finite Strains In the finite strain setting, we look to optimize the same actuator shown in Figure 4.1. Here, we model both the passive and responsive materials as compressible Neo-Hookean with strain energy functions
W˜l(F) = µl 2
kFk2F −2−2 log(DetF)+ 2µlνl
1−2νl (Det F −1)2, Wp(F) = µp
µl
W˜l(F),
(4.102)
where µl and µp are the shear moduli for the LCE and passive material. We use a Poisson ratio ofνl= 0.48 as to give a near-incompressable response while avoiding locking issues which may arise numerically from a fully incompressible material model. Similar to the small strain regime, we discretize the densitiesφ and ρas constant on each element. However, as we need to evaluate gradients of the director, we discretize this field through vector-valued first order finite elements. Thus, we consider the director field a as parameterized through its two scalar components
a(x) = a1(x)e1+a2(x)e2, (4.103)
where{e1, e2}is the standard basis inR2. Then, after updating these through MMA, we apply a radial return to a unit vector at each nodal point.
Figure 4.3 shows converged designs for optimal compliance ratio in (4.84) for varying shear moduli ratios of responsive to passive materials for a domain aspect ratio ofL/h= 2. This shares the general trends discussed for the small strain setting of Figure 4.2. However, they differ in that these designs use a large distribution of the responsive material along the bottom edge of the structure. Here, the finite deformation kinematics allow for the lengthening of these members to cause considerable vertical displacement.
4.4.2 3D Lifting Actuator
Small Strains In 3D, a parameterization through angles leads to a non- uniform discretization of the unit sphere. Thus, we consider the direction a parameterized through its 3 scalar components
a(x) = a1(x)e1+a2(x)e2+a3(x)e3 (4.104) where, again{ei}is the standard basis on R3.
We consider maximizing the blocking load of an actuator occupying a cantelev- ered rectangular prism under a uniform distributed load over a circular region on the far face shown in Figure 4.4. We update the director field through method of moving asymptotes, where we a apply a radial return to a unit vector at each element after updating. Again, we consider a Poisson ratio of ν = 0.48, and domain aspect ratio L/h = 2. Figure 4.5 shows converged designs computed on a 80×40×40 mesh. These designs are quite similar in nature to their 2D counterparts shown in 4.2.
Finite Strains In the finite strain setting, we look to optimize the same actuator shown in Figure 4.4. We consider both the passive and responsive materials as compressible Mooney-Rivlin material,
Wp(F) = µp µl
W˜l(F), (4.105)
where ˜Wl is the compressible Mooney-Rivlin model from (4.68). Here, we choose the α = 0.05 to give near Neo-Hookean response while maintaining coercivity. Again, we choose a Poisson ratio of νl = 0.48 and domain aspect
Figure 4.3: Converged designs for the rectangular lifting actuator of Figure 4.1 in the finite strain setting for varying stiffness ratios for the active vs passive material. The red indicates responsive material while blue denotes passive.
The left column shows the reference configuration, with director fields a is shown in black lines. The right column shows the deformed configuration after actuation.
Figure 4.4: 3D rectangular domain of side lengths L×h×h for the lifting actuator. A uniform, downward distributed load t is applied to a circular region at the center of the far face of the domain.
ratio L/h = 2. The director field a is parameterized through (4.104) and discretized using a first order vector-valued finite element space. Again, we update the director field through MMA, where we apply a radial return at each node after updates.
Figure 4.5 shows converged designs computed on a 80×40×40 mesh. While similar to the small-strain designs of 4.5, these have a smoother variation of the director field which arises from the penalty term.