Texture shape optimization analysis using a new acceleration gradient method based on the Taylor expansion and conjugate direction

Abstract

This paper proposes a new acceleration gradient method by addition of the Taylor expansion and conjugate direction to Nesterov’s acceleration gradient method. It was validated by updating the oil film thickness to minimize the friction coefficient on a textured surface. Nesterov’s acceleration gradient method converges faster than the gradient method for classical first-order optimization methods. The Taylor expansion of this mathematical technique reaches a more accurate approximation by incorporating higher-order terms. The conjugate direction is used in large-scale problems because it offers better convergence than the gradient descent method and is less memory-intensive than the Newton method. We introduced these into Nesterov’s acceleration gradient method to improve the convergence rate. The gradient of the design variable was obtained by the adjoint variable method. The results demonstrate the proposed method to converge faster than Nesterov’s accelerated gradient method. All the numerical calculations were performed by the finite element method using FreeFEM++.

Keywords

• texture
• shape optimization
• friction coefficient
• nesterov’s acceleration gradient method
• taylor expansion
• conjugate direction
• adjoint variable method
• finite element method