topopt-hw

Problem 12: Interior point method

In this assignment, we will use the classic MBB beam as an example to compare the performance differences of three different optimization methods: OC, MMA, and IP (Interior Point Method).

Results Comparison and Analysis

The volfrac is set to 0.5, penal is set to 3, and four groups are prepared with nelx=60, 120, 180, and 240, nely=nelx/3, and rmin=nelx*0.04.

This image corresponds to the optimization results and objective function variation curves for different methods, with nelx=60, 120, 180, and 240 from top to bottom, and OC, MMA, and IP from left to right.

This image illustrates the variation of the objective function value with the increase in the number of elements.

This image illustrates the variation in runtime as the number of elements increases.

Based on the two plots, we can analyze the performance of the three methods — OC (Optimality Criteria), MMA (Method of Moving Asymptotes), and IP (Interior Point Method) — in terms of compliance and time across different element sizes.

1. Compliance vs. No. of Elements:

In the first plot, which shows the relationship between compliance and the number of elements:

• OC and MMA exhibit very similar behavior, with both methods having a nearly flat curve. The compliance values increase only slightly as the number of elements grows from 60x20 to 240x80, indicating that both methods maintain a relatively consistent performance in terms of compliance, regardless of mesh density.
• IP shows a significantly different trend. Its compliance values increase sharply as the number of elements increases, particularly between 120x40 and 240x80. This suggests that the IP method becomes less efficient at minimizing compliance as the problem’s complexity increases.

2. Time vs. No. of Elements:

The second plot shows the computational time required by each method:

• MMA performs well initially but shows a dramatic increase in computation time as the number of elements grows. Between 180x60 and 240x80, the time jumps significantly, making MMA the slowest method for the largest problem size.
• OC also shows an increase in time, but it performs better than MMA, especially for larger element counts. OC remains more computationally efficient than MMA when handling larger problems.
• IP, on the other hand, shows almost no significant increase in time as the number of elements grows. Its time remains relatively constant, indicating a much more stable and scalable computational cost compared to both OC and MMA.

Summary:

• Compliance: Both OC and MMA offer stable compliance performance, with minimal increase in compliance as the number of elements grows. IP, however, suffers from a significant increase in compliance with larger problem sizes, making it less effective in this regard.
• Time: IP is the most time-efficient method, showing almost no increase in computational time as problem size increases. OC performs moderately well, with a noticeable but manageable increase in time for larger problems. MMA, however, is the least time-efficient method, showing a significant increase in time as the number of elements increases.

In conclusion, IP is the most computationally efficient method in terms of time, but its compliance performance degrades as problem size increases. OC provides a good balance between compliance and computational efficiency, making it a reliable choice for larger problems. MMA delivers good compliance but is significantly slower for larger problem sizes, making it less suitable for large-scale problems.