Optimizing Design of Components in Unilateral Contact with Pressure Constraints Using Gradient-Based and Gradient-Free Methods

Core Concepts

This paper presents gradient-based and gradient-free optimization methods to solve design optimization problems involving deformable bodies in unilateral contact, subject to pressure constraints.

Abstract

The paper addresses the challenge of combining state-of-the-art contact finite element solvers with optimization algorithms for design optimization of contacting systems. Two main approaches are explored:

Gradient-based optimization:

The authors derive and implement sensitivities for design optimization problems with pressure objectives and constraints.
They use the interior-point solver Ipopt to solve the optimization problems.
The gradient-based approach is able to handle the nonsmoothness of the contact problems by carefully designing the bound constraints.

Gradient-free optimization:

The authors design and apply constrained Bayesian optimization with Gaussian Process surrogate models.
Bayesian optimization treats the objective and constraint functions as black-box and does not require sensitivity information.
The constrained Bayesian optimization algorithm is able to find improved designs, though the accuracy is reduced compared to the gradient-based approach.

The authors present two numerical examples inspired by real-life engineering applications, such as the design of high-current joints and Marman clamps, to demonstrate the effectiveness, strengths and limitations of both optimization methods for problems with changing contact regions and pressure constraints.

Stats

The elastic modulus E is set to 200 and Poisson's ratio ν is 0.3 for the wedge problem.
The elastic modulus E is set to 2 × 10^6 and Poisson's ratio ν is 0.3 for the Marman clamp problem.

Quotes

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Key Insights Distilled From

Design optimization in unilateral contact using pressure constraints and Bayesian optimization

by Jingyi Wang,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03081.pdf

Deeper Inquiries

How can the nonsmoothness of the contact problems be addressed more directly to develop more robust and efficient optimization algorithms?

In order to address the nonsmoothness of contact problems more directly and develop more robust and efficient optimization algorithms, several strategies can be employed:

Improved Sensitivity Analysis: Enhancing the sensitivity analysis techniques to accurately capture the changes in the objective and constraint functions with respect to the design variables. This can help in better understanding the impact of design changes on the contact behavior and pressure distribution.

Advanced Surrogate Models: Utilizing more advanced surrogate models, such as neural networks or deep learning models, to approximate the complex and nonlinear behavior of the contact problems. These models can better handle nonsmooth functions and provide more accurate predictions.

Incorporating Physics-Informed Models: Integrating physics-based models into the optimization process can help in capturing the underlying physical phenomena of contact problems. By incorporating domain knowledge, the optimization algorithms can better navigate the nonsmooth regions.

Adaptive Mesh Refinement: Implementing adaptive mesh refinement techniques to refine the mesh in regions of interest, such as contact surfaces, can improve the accuracy of the contact simulations and optimization results.

Hybrid Optimization Approaches: Combining gradient-based and gradient-free optimization methods in a hybrid approach can leverage the strengths of both methods to handle nonsmoothness more effectively. This can involve using gradient information where available and switching to gradient-free methods in challenging regions.

By implementing these strategies, optimization algorithms can better handle the nonsmooth nature of contact problems, leading to more robust and efficient solutions.

How would the performance of the gradient-based and gradient-free methods scale with the problem size and dimensionality of the design space?

The performance of gradient-based and gradient-free methods can vary based on the problem size and dimensionality of the design space:

Gradient-Based Methods:

Scaling with Problem Size: Gradient-based methods, such as interior-point optimization, typically scale well with problem size, especially for problems with a large number of design variables. They can efficiently handle high-dimensional spaces and large-scale optimization problems.
Sensitivity to Dimensionality: However, as the dimensionality of the design space increases, gradient-based methods may face challenges in computing accurate gradients and sensitivities, especially in nonsmooth regions. This can lead to convergence issues and slower performance in high-dimensional spaces.

Gradient-Free Methods:

Scaling with Problem Size: Gradient-free methods, like Bayesian optimization, can be computationally expensive for large-scale problems due to the need for multiple function evaluations. As the problem size increases, the computational cost of evaluating the objective function at each iteration can become prohibitive.
Robustness in High Dimensions: Gradient-free methods are generally more robust in high-dimensional spaces where computing gradients is challenging. They can handle complex, nonsmooth functions without relying on gradient information, making them suitable for problems with high dimensionality.

In summary, while gradient-based methods are efficient for large-scale problems but may struggle with high-dimensional spaces, gradient-free methods offer robustness in handling nonsmooth functions but can be computationally demanding for large-scale and high-dimensional problems.

What other real-world engineering applications could benefit from the optimization approaches presented in this paper?

The optimization approaches presented in the paper can benefit various real-world engineering applications, including:

Automotive Design: Optimizing the shape and material distribution of automotive components to improve crashworthiness and structural integrity while considering contact constraints.

Aerospace Engineering: Designing aircraft components, such as wing structures or landing gear, to optimize weight, strength, and contact behavior under different loading conditions.

Biomechanical Engineering: Optimizing the design of prosthetics or orthopedic implants to ensure proper contact pressure distribution and biomechanical performance.

Manufacturing Processes: Optimizing the design of tooling and fixtures in manufacturing processes to improve efficiency, reduce wear, and ensure proper contact between components.

Civil Engineering: Designing structural elements in buildings and bridges to optimize load distribution, contact behavior, and overall structural performance.

By applying the optimization approaches to these and other engineering applications, it is possible to enhance design efficiency, performance, and reliability while considering complex contact constraints.