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A multi-material level set-based topology optimization of flexoelectric composites
Hamid Ghasemi, Harold S. Park, Timon Rabczuk
TL;DR
Single-material flexoelectric sensors and actuators might be suboptimal, motivating multi-material topology optimization. The paper develops a level set-based design approach and demonstrates it on two-, three-, and four-phase composites.
Problem
Single-material flexoelectric sensors and actuators might be suboptimal, while prior topology-optimization approaches leave simultaneous multi-material design insufficiently addressed.
Method
The methodology uses a multi-phase vector level set model, B-spline elements, and Hamilton-Jacobi updates for simultaneous topology optimization of flexoelectric composites.
Results
Numerical examples demonstrate the model's capability to design two-, three-, and four-phase flexoelectric composites.
Takeaways & Limitations
The results support multi-material topology optimization as a flexible framework for designing flexoelectric composites with active, passive, and hole regions.
Takeaways & Limitations
Future work is needed on numerical stability and updating procedures because level set functions can become too flat or too steep, causing convergence issues.
Abstract
from arXiv · showhide
We present a computational design methodology for topology optimization of multi-material-based flexoelectric composites. The methodology extends our recently proposed design methodology for a single flexoelectric material. We adopt the multi-phase vector level set (LS) model which easily copes with various numbers of phases, efficiently satisfies multiple constraints and intrinsically avoids overlap or vacuum among different phases. We extend the point wise density mapping technique for multi-material design and use the B-spline elements to discretize the partial differential equations (PDEs) of flexoelectricity. The dependence of the objective function on the design variables is incorporated using the adjoint technique.The obtained design sensitivities are used in the Hamilton Jacobi equation to update the LS function. We provide numerical examples for two, three and four phase flexoelectric composites to demonstrate the flexibility of the model as well as the significant enhancement in electromechanical coupling coefficient that can be obtained using multi-material topology optimization for flexoelectric composites.
1. Introduction
Flexoelectric sensors and actuators offer useful structural and material advantages but face efficiency, brittleness, and performance tradeoffs. The paper addresses these limitations by optimizing multi-phase flexoelectric composites with a level set framework.
- Motivation: Flexoelectric sensors and actuators support applications including biomedical systems, environmental monitoring, and structural health monitoring.
- Motivation: Their structural simplicity, high power density, and broad material choice are offset by usually low efficiency.
- Challenges: Ceramic and single-crystal flexoelectric materials are typically brittle, whereas polymers are flexible but have weaker flexoelectric performance.
- Challenges: High-strain-gradient zones contribute more to electrical energy generation, making single-material flexoelectric structures potentially suboptimal.
- Research opportunity: Multi-phase flexoelectric composites offer an opportunity to bridge high flexoelectric performance and poor structural properties.
- Contribution: The paper uses level set optimization to simultaneously design elastic, flexoelectric, and void phases in multi-material sensors and actuators.
- Research gap: Previous multi-material topology-optimization studies were relatively rare, mostly used SIMP, and had not examined multi-material flexoelectric composites.
2. A summary of the governing equations and discretization
The paper formulates flexoelectricity through governing equations and a weak form, then discretizes the mechanical and electrical fields with B-spline basis functions. The resulting discrete system incorporates material tensors, spatial derivatives, and boundary conditions.
- Governing equations: The flexoelectric formulation defines usual, higher-order, and physical stresses and electric displacements through constitutive relations.
- Weak form: The governing equations are converted into a weak form using Hamilton’s principle after imposing boundary conditions and integrating over the domain.
- Weak form: In the weak form, u_i denotes mechanical displacement, θ electric potential, t̅_i prescribed traction, and ϛ surface charge density.
- Discretization: B-spline basis functions approximate the mechanical displacement and electric-potential fields using nodal parameters at mesh control points.
- Discretization: The discrete system assembles element integrals over Ω_e, with B_u and B_θ containing spatial derivatives of the B-spline basis functions.
- Material representation: The formulation uses second derivatives through H_u and represents C, κ, e, and μ in matrix form for elasticity, dielectric, piezoelectric, and flexoelectric behavior.
3. Level Set Method (LSM) and optimization problem
The method represents multiple material phases with vector level-set functions, maps them to element densities, and updates boundaries through sensitivity-driven Hamilton–Jacobi evolution. The optimization combines flexoelectric electromechanical objectives with volume constraints and iterative numerical solution procedures.
- Density mapping and discretization: B-spline basis functions discretize the level-set and flexoelectric PDE representations through control-point design variables.Each level-set function is associated with design variables defined on a mesh of control points.
- Level-set evolution: The zero iso-surface implicitly represents each design boundary, while Hamilton–Jacobi evolution updates the level-set function using sensitivity-based velocity fields.The level-set function is initialized as a signed-distance function and advanced with an explicit first-order upwind scheme.
- Multi-phase level-set representation: Vector level sets partition the design domain into n = 2^m regions, allowing each point to belong to exactly one material phase.This formulation addresses partition-condition and high-level-set-function challenges in conventional multi-phase partitioning.
- Multi-phase level-set representation: For three phases including void, Φ1 separates solid from void while Φ2 distinguishes the solid material phases.Two level-set functions can represent four material phases through different combinations of their regions.
- Density mapping and discretization: Point-wise Heaviside mapping assigns element-wise phase densities from level-set values at finite-element centers.The resulting densities satisfy 0 ≤ ρk ≤ 1 and are embedded in the electromechanical problem to obtain effective material properties.
- Optimization problem: The objective uses electrical and mechanical energies to define the electromechanical coupling coefficient while augmented Lagrangian terms enforce phase-volume constraints.The constrained problem is replaced by sequential unconstrained subproblems with updated Lagrangian multipliers until convergence.
4. Numerical examples
Numerical examples evaluate two-, three-, and four-phase composite beams under specified mechanical and electrical conditions. Multi-material combinations enhance electromechanical coupling, while phase fractions, topology, and aspect ratio determine performance.
- Experimental setup: The examples study two-, three-, and four-phase composite beams using quadratic B-spline discretization under plane-strain conditions.The standard beam is 60 × 15 μm, discretized by 48 × 12 quadratic B-spline elements, loaded by 100 μN, with open-circuit electrical boundaries.
- Two-phase composites: In Case-1, combining active and passive phases produces a higher normalized electromechanical coupling coefficient than the single-phase counterpart, with an optimal composition.Increasing soft passive material raises mechanical participation but can reduce the electromechanical coupling contribution, creating a tradeoff.
- Two-phase composites: Any combination of Active 1 and Active 2 yields higher k2 than either single-phase Active 1 or Active 2.This result concerns Case-2, where the active phases are combined in a two-phase composite.
- Three-phase composites: The optimal three-phase topology places flexoelectric material around the perimeter and elastic material in the interior while maintaining the void fraction.As the flexoelectric volume fraction decreases, the elastic phase increases and the void phase remains constant at 0.3V0.
- Three-phase composites: The phase volumes and objective function converge smoothly and precisely toward their specified constraints during optimization.The convergence behavior is illustrated for the design with flexoelectric, elastic, and void volumes [0.14, 0.56, 0.3] × V0.
- Phase-fraction effects: 0.00022 (k_n^2 = 5.57) is obtained with 14% flexoelectric phase, while 0.00037 (k_n^2 = 9.14) is obtained with 28%.Further increasing the flexoelectric fraction decreases coupling, reaching k_n^2 = 3.98 for the 70% flexoelectric and 30% void design.
- Aspect-ratio effects: The flexoelectric size effect and flexoelectric volume ratio have contradictory effects on k2 across beam aspect ratios.The highest k2 occurs for aspect ratio 8 despite less active material, whereas the volume ratio makes aspect ratio 6 underperform aspect ratio 4.
5. Concluding remarks
The methodology combines multi-phase vector level sets, pointwise density mapping, B-spline discretization, and adjoint sensitivities to design two-, three-, and four-phase flexoelectric composites. Numerical examples show enhanced electromechanical coupling, while revealing a tradeoff between electrical and mechanical energy effects and identifying numerical-stability issues for future work.
- B-spline elements were successfully implemented to model the flexoelectric effect in the composite designs.
- The model combines flexoelectric and dielectric materials with a vector level set technique to enhance electromechanical performance in multiphase sensors and actuators.
- The numerical examples demonstrate designs for two-, three-, and four-phase composites with an optimal electromechanical coupling coefficient defined by k2.
- At optimal constituent volume fractions, the two-phase active-passive composite achieves a normalized electromechanical coupling coefficient larger than that of a beam made purely from the active material.
- For the three-phase active-passive-void composite, k_n^2 increases by a factor of 9.
- Increasing passive-material volume fraction decreases k2 through greater mechanical energy but increases it through higher strain gradients and electrical energy, producing an optimal tradeoff.
- Future work will study numerical stability, the updating procedure, geometry mapping, regularization, and convergence issues associated with overly flat or steep level-set functions.
Appendix A: Sensitivity analysis
The appendix formulates the coupled flexoelectric PDEs as a global residual system and derives objective and volume-constraint sensitivities with respect to level-set variables. It applies chain-rule and adjoint-style differentiation for multi-material phase cases.
- The coupled displacement and electric-potential equations are expressed in a single global residual form.
- The objective function and solid and flexoelectric volume constraints are differentiated with respect to the level-set variables using the chain rule.
- The chain-rule sensitivity includes a term obtained by differentiating the global residual equation.
- Material-property derivative terms are obtained according to Eq. (19), and the appendix provides separate expressions for four- and three-material-phase cases.
- The residual formulation includes a zero-valued branch for conditions specified in Eq. (A14).