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A Comparative Study on Robust Topology Optimization of Design-Dependent Pressure-Actuated Compliant Mechanisms with Quadrilateral Elements
Swagatam Islam Sarkar, Prabhat Kumar
TL;DR
Q4 meshes can produce checkerboards and point connections, while design-dependent loads change as optimization progresses. This paper compares robust topology-optimized pressure-actuated mechanisms using Q4, Q8, and Q9 elements. Q8 yields the better inverter objective, whereas Q4 yields the better gripper objective.
Problem
Q4-based topology optimization can produce checkerboards and point connections, and design-dependent loads change direction, location, or magnitude during optimization.
Source: §1 · p. 2Method
Results
Q8 gives the better objective value for inverters, while Q4 gives the better objective value for the gripper.
Source: §6 · p. 10Takeaways & Limitations
The element choice corresponds to different preferred objective results across the two mechanisms, and the optimized topologies differ among quadrilateral elements.
Source: §5 · pp. 7, 9
Abstract
from arXiv · showhide
This paper presents a comparative study of compliant mechanisms generated using a robust topology optimization technique involving design-dependent pressure loads. Design domains are parameterized using standard and higher-order quadrilateral elements. Both eroded and blueprint configurations are considered. A min-max optimization model combined with an output-spring method is employed to extremize the mechanisms' output displacements. A volume and a strain energy constraint are applied to the blueprint and the eroded designs, respectively. The optimization process is executed using the method of moving asymptotes. Numerical experiments are performed to optimize the pressure-actuated inverter and gripper mechanisms using Q4, Q8, and Q9 elements, and the results are compared. The research highlights how quadrilateral element selection influences both the resulting topologies and performance characteristics.
1 Introduction
Topology optimization distributes material to meet a design objective, but Q4 meshes can produce checkerboards and point-connected features, while design-dependent loads change during optimization. This study compares robust designs of pressure-actuated compliant mechanisms using quadrilateral elements and adopts Darcy’s method to model the changing pressure loads.
1 Introduction
- The study compares density-based topology optimization of compliant mechanisms using standard and higher-order quadrilateral elements under design-dependent loading in a robust framework.The design-dependent pressure loads are modeled using the Darcy method.Source: §1 · pp. 1–2
- Q4 elements are widely used for their simplicity, but optimized solutions can exhibit checkerboards and point connections; filtering, robust formulations, and higher-order quadrilaterals are among the proposed remedies.The higher-order elements discussed are 8-node and 9-node quadrilaterals.Source: §1 · p. 2
- Design-dependent loads change direction, location, or magnitude as optimization progresses, and prior work has addressed them in compliant mechanisms using approaches including Darcy’s law.This study adopts the Darcy method to model design-dependent pressure loads.Source: §1 · p. 2
- Robust formulation addresses manufacturing-tolerance effects by distinguishing the worst-case over-etched eroded design from the intended blueprint design.Prior work applied this framework to pressure-actuated compliant mechanisms.Source: §1 · p. 2
2 Quadrilateral Finite Elements
The paper discretizes design domains with Q4, Q8, and Q9 quadrilateral finite elements and compares the resulting optimized designs. It notes that using more nodes per element generally improves accuracy.
2 Quadrilateral Finite Elements
- The design domains are discretized with four-node Q4, eight-node Q8, and nine-node Q9 quadrilateral elements, whose optimized results are compared.The paper notes that more nodes per mesh element generally improve accuracy.Source: §2 · p. 3
- Figure 1 depicts Q4, Q8, and Q9 elements, with node numbers and the ξ-η natural coordinate system indicated.The shape functions are evaluated according to node numbers and positions.Source: Figure 1 · p. 3, §2 · p. 3
3 Pressure load modelling
The pressure-load model uses Darcy’s law with a drainage term to represent pressure behavior at the solid-void interface. In the finite-element model, the pressure field produces nodal forces that cause deformation.
3 Pressure load modelling
- Darcy’s law relates flux to the pressure gradient, with permeability and fluid viscosity determining the flow coefficient K.The paper identifies k as permeability and µ as viscosity.Source: §3 · pp. 3–4
- The model defines the physical design variable and a smooth Heaviside function with tuning parameters for its step location and slope.These quantities enter the flow-coefficient description for void and solid elements.Source: §3 · p. 4
- The model adds a drainage term based on the drainage coefficient and the difference between continuous and external pressure to represent the solid-void interface.The drainage term is combined with Darcy’s law in the equilibrium equation.Source: §3 · p. 4
- The flow equation is written as AP = 0 in finite-element form, and the resulting pressure field gives nodal forces that cause deformation.A and P denote the global flow matrix and global pressure field; T, F, K, and U denote the global transformation matrix, force vector, stiffness matrix, and displacement vector.Source: §3 · p. 4
4 Topology optimization for robust design
The robust design uses density-based topology optimization with modified SIMP stiffness interpolation, then minimizes the worst of the blueprint and eroded output displacements under volume and strain-energy constraints.
4 Topology optimization for robust design
- Modified SIMP interpolates element stiffness from void and solid phases using the projected physical design variable, with penalization factor p = 3 and Ev = 10^-6Es.The physical design variable is obtained through a Heaviside projection filter.Source: §4 · p. 4
- The formulation is applied to the inverter and gripper compliant-mechanism design problems.The robust topology-optimization formulation cited as is employed for pressure-actuated compliant mechanisms.Source: §4 · p. 4
- The robust formulation minimizes the maximum of the blueprint and eroded output displacements, using an output spring with stiffness ks = 1 at the output degree of freedom.The virtual load vector selects the desired output degree of freedom to evaluate each displacement.Source: §4 · p. 5
- The blueprint design is subject to a volume constraint, while the eroded design is subject to a strain-energy constraint.The method of moving asymptotes updates the design variables, and sensitivities are evaluated using the adjoint-variable method.Source: §4 · p. 5
5 Results and Discussions
Results compare pressure-loaded inverter and gripper designs across Q4, Q8, and Q9, including blueprint and eroded configurations; the preferred element varies by mechanism and configuration.
5 Results and Discussions
- At Se/S*=0.95, Q8 gives the inverter blueprint better output displacement, Q4 favors its eroded design, while Q8 is more flexible for both configurations at 0.80.Source: §5 · pp. 6–7, Table 1 · p. 7, Table 2 · p. 8
- For the inverter at Se/S*=0.95, blueprint volume reaches its 0.25 limit and eroded volume is 0.227 for Q4, Q8, and Q9.Source: Table 1 · p. 7, §5 · p. 7
- Optimized topologies vary by element: inverter thickness differs across Q4, Q8, and Q9, and the gripper's Q4 topology differs substantially from the others.Source: §5 · pp. 7, 9, Table 2 · p. 8, Table 3 · p. 9
- The inverter objective f0 shows convergent behavior in plots for strain-energy fractions 0.95 and 0.80.Source: §5 · p. 7, Figure 3 · p. 8
- For the gripper, Q4 produces better objective values than Q8 and Q9 for both blueprint and eroded designs.Source: Table 3 · p. 9, §5 · p. 9
- Across the reported cases, eroded mechanisms are described as more flexible, while blueprint mechanisms are identified as more practical to realize.Source: §5 · p. 9
6 Conclusions
The study compares robust pressure-actuated inverter and gripper mechanisms optimized with Q4, Q8, and Q9 elements. Element choice is associated with differences in topology and performance, with Q8 favoring the inverter and Q4 the gripper.
6 Conclusions
- The comparison covers pressure-actuated inverter and gripper mechanisms optimized with Q4, Q8, and Q9 quadrilateral elements.Source: §6 · p. 10
- The robust formulation uses a min-max objective over blueprint and eroded output deformations, with volume and strain-energy constraints on those respective designs.Source: §6 · p. 10
- Q8 suggests more inverter output displacement, whereas Q4 yields a better objective value for the gripper.Source: §6 · p. 10
- Quadrilateral element selection produces slight variations in optimized results and affects both topology and performance; the gripper's eroded design is more flexible at the output than its blueprint.Source: §6 · p. 10