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Proportional Topology Optimization: A new non-gradient method for solving stress constrained and minimum compliance problems and its implementation in MATLAB
Emre Biyikli, Albert C. To
TL;DR
Stress-constrained topology optimization is more challenging than compliance optimization, and gradient calculations can add analytical and computational burden. PTO provides a non-gradient approach; the reported comparisons show similar PTOc results to Top88 and trade-offs between PTOs and PTOc.
Problem
Stress-constrained topology optimization is challenging because it has high nonlinearity, and stress gradients can be analytically complicated and computationally burdensome.
Source: Introduction · p. 2Method
PTO assigns element design variables proportionally to stress or compliance and manages them through global constraints without gradients.
Source: Introduction · p. 3Results
PTOc attains very similar results to Top88 with the same efficiency; cross-fed comparisons report lower stress and volume for PTOs but higher compliance than PTOc.
Source: Numerical examples · p. 9, Cantilever · p. 12, Conclusions · p. 13Takeaways & Limitations
The authors present PTO as a simple, efficient, and accurate non-gradient method and provide self-contained MATLAB programs for both problem types.
Source: Introduction · p. 3The authors call for more comprehensive parameter studies and further investigation of mesh dependency; the current method treats the whole domain as one cluster.
Source: Conclusions · p. 13
Abstract
from arXiv · showhide
A new topology optimization method called the Proportional Topology Optimization (PTO) is presented. As a non-gradient method, PTO is simple to understand, easy to implement, and is also efficient and accurate at the same time. It is implemented into two MATLAB programs to solve the stress constrained and minimum compliance problems. Descriptions of the algorithm and computer programs are provided in detail. The method is applied to solve three numerical examples for both types of problems. The method shows comparable efficiency and accuracy with an existing gradient optimality criteria method. Also, the PTO stress constrained algorithm and minimum compliance algorithm are compared by feeding output from one algorithm to the other in an alternative manner, where the former yields lower maximum stress and volume fraction but higher compliance compared to the latter. Advantages and disadvantages of the proposed method and future works are discussed. The computer programs are self-contained and publicly shared in the website www.ptomethod.org.
Introduction
The introduction frames topology optimization as material removal for stronger, lighter designs and distinguishes stress-constrained from minimum-compliance problems. It motivates PTO as a simple non-gradient method and presents its MATLAB implementations.
Introduction
- Topology optimization systematically removes redundant material to seek higher strength-to-weight designs.Source: Introduction · p. 2
- The stress problem minimizes volume fraction subject to stress constraints, while the compliance problem minimizes compliance for a given volume fraction.The paper treats an element as a finite element and notes that optimized geometries require post-processing.Source: Introduction · p. 2
- Stress optimization is challenging because of high nonlinearity, and its gradient calculations can be analytically complicated and computationally burdensome.The introduction describes a trade-off between gradient methods’ efficiency and non-gradient methods’ lower implementation complexity.Source: Introduction · pp. 2–3
- Prior approaches include OC, SLP, SQP, CONLIN, MMA, and ESO; OC assigns design variables proportionally to objective-function values.ESO removes elements based on objective values, while BESO also adds them.Source: Introduction · p. 2
- PTO assigns element design variables proportionally to stress or compliance, imposes constraints globally, and uses a heuristic non-gradient search.Continuous density variables are described as improving search performance and preserving intermediate-density flexibility.Source: Introduction · p. 3
- The authors provide self-contained MATLAB programs for stress and compliance problems and freely share them for educational and research use.They describe PTOs as, to their knowledge, the first publicly shared self-contained program for continuum stress-constrained optimization.Source: Introduction · p. 3
Stress constrained problem
The stress problem minimizes volume fraction subject to stress constraints, while the compliance problem minimizes compliance under a volume-fraction constraint. PTO uses separate stress- and compliance-based algorithms to distribute material and update element densities.
Stress constrained problem
- The stress problem minimizes volume fraction while satisfying stress constraints, whereas the compliance problem minimizes compliance subject to a volume-fraction constraint.The stress formulation bounds elemental densities and stress; the compliance formulation adds compliance and total mass to the problem’s nomenclature.Source: Stress constrained problem · p. 3, Minimum compliance problem · p. 3
The PTO algorithms
- PTOs increases or decreases target material according to whether maximum stress exceeds the allowable limit, then distributes material in proportion to elemental stress.An inner loop reapportions material after filtering and density limits until the remaining amount is small; the next main-loop update blends the previous and optimized densities.Source: The PTO algorithms · p. 4, Figure 1 · p. 4
- PTOc fixes its target material amount from the start and distributes it in proportion to elemental compliance rather than stress.It recalculates elemental compliance each iteration and stops when the maximum density change between successive iterations is below 0.01.Source: The PTO algorithms · p. 5, Figure 2 · p. 5
Material model
PTO uses a modified SIMP density approach to improve search performance while maintaining near 0/1 solutions.
Material model
- The modified SIMP density approach is adopted to improve search performance while maintaining near 0/1 solutions.It defines density-dependent Young’s modulus using a small void modulus and a penalized solid-material modulus.Source: Material model · p. 5
- The modified SIMP model assigns void elements a typically 10^-9 Young’s modulus and uses a typically 3 penalty coefficient.The relation connects density-dependent Young’s modulus to the void and solid-material moduli.Source: Material model · p. 5
- Because the small void modulus already avoids stiffness singularities, the modified SIMP approach makes a lower density bound redundant.The passage also states that the approach drives densities toward 0 and 1 because volume varies linearly while stiffness varies with the penalty power.Source: Material model · p. 5
Stress constraint
PTO constrains maximum elemental stress using von Mises stress at each element’s geometric center.
Stress constraint
- The maximum stress measure constrains stress at every element, giving tight control because no element may exceed the elastic limit.The stress value is the von Mises stress at the element’s geometric center.Source: Stress constraint · p. 5
- The maximum stress function is nondifferentiable, so it cannot be used with gradient methods.Gradient methods usually use a p-norm instead, which is less tight unless p is very large; at p = ∞ it is equivalent to maximum stress.Source: Stress constraint · p. 5
- Unlike the p-norm measure, maximum stress has physical meaning and is the simplest stress measure to implement.The passage notes that the p-norm is not as tight unless its exponent is very large.Source: Stress constraint · p. 5
Density filtering
PTO uses volume-preserving density filtering that locally averages densities and smooths the design.
Density filtering
- The cone density filter locally averages neighboring densities with weights inversely proportional to their distance and preserves volume.The results show filtered densities.Source: Density filtering · p. 5
- Filtering prevents small-scale features such as jagged edges, narrow members, and sharp interfaces, and prevents checkerboards [Le et al. 2010; Sigmund 2007].Smoothing also produces a blurred region around structural members [Le et al. 2010].Source: Density filtering · pp. 5–6
- Filtering is described as helping the algorithm avoid local minima and imposing a minimum design length scale [Le et al. 2010; Sigmund 2007].The cited passage lists these as advantages of filtering.Source: Density filtering · p. 6
- Filtering is said to ensure solution existence, although that claim is not proven [Sigmund 2001].The passage explicitly qualifies this stated advantage as unproven.Source: Density filtering · p. 6
Control parameters
PTO tunes how stress or compliance values determine material distribution, with separate settings for stress and compliance problems. The section also specifies boundary-condition handling and the stress measure used in results.
Control parameters
- The proportion exponent controls how strongly elemental density allocation follows stress or compliance, while the history coefficient blends prior and newly optimized densities.A history coefficient of 0.5 blends the two equally; the preliminary study reports proportion exponents of 2.0 for stress and 1.0 for compliance, and history coefficients of 0.0 and 0.5, respectively.Source: Control parameters · p. 6
Boundary conditions
- Distribute boundary conditions over sufficient elements: excluding boundary-adjacent elements can change solutions and appear to improve results, while imposing conditions on too few can cause poor behavior from stress or compliance concentration.If the resulting structure seems fragile under loading, material may be added near loading regions during post-processing.Source: Boundary conditions · p. 6
Stress measure
- Stress results refer to von Mises stress measured at element geometric centers for 2-D plane-stress examples using bilinear square elements.The passages define the 2-D stress tensor using D, B, and u, and identify D as the constitutive matrix, B as the shape-function derivative matrix, and u as the displacement vector.Source: Stress measure · pp. 6–7
MATLAB programs
The MATLAB programs adapt Top88’s finite-element setup for stress-constrained and compliance topology optimization, with distinct PTO update rules. They also expose input controls and describe implementation, verification, and extension options.
Program design
- The two MBB beam programs inherit much of Top88’s finite-element setup and solution, while replacing its optimization algorithm and adding stress analysis.They also remove sensitivity analysis and add inputs for element edge length, load distribution, and density bounds.Source: MATLAB programs · p. 7
Implementation
- The programs build a density filter from distance-weighted neighboring elements, then each main-loop iteration solves the finite-element system and computes elemental stress and compliance.The reported stress values are actual, not density-normalized or stress norms, and the programs were verified against ANSYS by comparing displacement, compliance, and stress.Source: MATLAB programs · pp. 8–9
Stress program (PTOs)
- PTOs adjusts target material according to whether maximum elemental stress exceeds its limit, then iteratively distributes remaining material in proportion to elemental stress raised to exponent q.Filtering and density bounds are applied during distribution, which repeats until the remaining material is at most 0.001; termination also requires the maximum stress to approach its limit within 0.001 and more than 50 iterations.Source: MATLAB programs · p. 8
Compliance program (PTOc)
- PTOc sets target material from the volume-fraction constraint and distributes it in proportion to elemental compliance, while blending prior and current densities using history coefficient α.It stops when the maximum density change is below 0.01 after more than 50 iterations; unlike PTOs, its volume constraint is satisfied a priori.Source: MATLAB programs · pp. 8–9
Use and extensions
- The programs accept normalized volume and density bounds and can be extended to prescribe solid or void regions or to impose both stress and compliance constraints.Consistent physical units can be attached; element-count parameters such as load distribution and filter radius are in units of elements.Source: MATLAB programs · p. 9
Numerical examples
Across three numerical examples, PTOs and PTOc produce near black-and-white topologies, with a trade-off between compliance and volume at matched stress levels. The examples use common material, filtering, and loading settings, and include MBB, cantilever, and L-bracket designs.
Common settings
- All examples use E0=1, ν=0.3, Emin=10-9, penal=3, a unit load distributed over three elements, and element edge length 1.The density bounds are 0 and 1, the filter radius is 1.5, and PTOs uses q=2 while PTOc uses α=0.5.Source: Numerical examples · p. 9
Example setup
- The three examples are the right half of an MBB beam, a cantilever beam, and an L bracket, with distinct discretizations, supports, and load placements.The MBB and cantilever meshes are 120x40 and 120x60 elements; the L bracket is 100x40 elements along its long and short edges.Source: Numerical examples · p. 9
Topology contrast
- Average contrast indices are 0.83 for PTOc and 0.85 for PTOs, respectively, and these indices indicate that both methods produce near black-and-white topologies.Source: Numerical examples · p. 10
PTOs and PTOc comparison
- 6.3% less compliance for PTOc solutions on average, while PTOs solutions use 7.3% less volume; across all six cases, PTOs has higher compliance but lower volume fraction at the same stress level.The topologies are nearly identical for the cantilever, but differ considerably for the MBB beam and L bracket; PTOc tends toward thicker members and PTOs toward more members.Source: Numerical examples · p. 9, Table 1 · p. 10
Cantilever
Across three examples, PTOs improves the stress–volume trade-off over PTOc, while PTOc’s compliance curves closely match Top88. Topology similarities and differences vary by example.
Cantilever
- For the same stress level, PTOs has higher compliance but lower volume fraction than PTOc in all six cases.PTOc tends to have thicker structural members, while PTOs tends toward a greater number of members.Source: Figure 5 · p. 11
- PTOc and Top88 have indistinguishable compliance-versus-volume-fraction curves across the MBB beam, cantilever beam, and L bracket.PTOc takes 26.6% more, 0.4% less, and 23.7% less iterations than Top88 in the examples’ presented order.Source: Cantilever · p. 11, Figure 6 · p. 12
- PTOc and Top88 topologies are similar for the cantilever but markedly different for the MBB beam and L bracket.At volume fraction 0.35, PTOc’s L-bracket topology contains a tiny feature near the loading that the authors describe as fragile against transverse loading.Source: Cantilever · p. 11, Figure 7 · p. 12
- On average, PTOs provides 8.4% less stress for the same volume fraction and 5.9% less volume for the same stress than PTOc.The comparison covers the MBB beam, cantilever beam, and L bracket; improvement extent depends on the example.Source: Cantilever · p. 12, Table 2 · p. 12, Figure 8 · p. 13
Conclusions
PTO is a simple non-gradient method that the authors report achieves considerable efficiency and accuracy; PTOc produces results similar to gradient-based Top88. The authors identify mesh-dependency and further method development as open work.
Conclusions
- PTO is a non-gradient method that avoids gradient derivations and implementation, trading weaker mathematical rigor for simplicity.The authors describe its efficiency and accuracy as considerable given its simplicity.Source: Conclusions · p. 13
- PTOc attains results very similar to Top88 without gradients while maintaining the same level of efficiency.The authors say PTOs has not been shown here to match state-of-the-art methods for continuum stress problems, especially gradient-based methods, and leave comparison for future work.Source: Conclusions · p. 13
- The authors present standalone MATLAB programs for stress and compliance problems as potential tools for education, industry, and research.The programs solve the MBB beam example and are publicly shared.Source: Conclusions · p. 13
- Quantitative comparisons are still needed to assess mesh independence, which the authors say is currently supported only by topology comparisons.They also call for more comprehensive parametric work and further investigation of element clustering.Source: Conclusions · p. 13
Appendix A – PTOs
The two MATLAB functions share finite-element setup, loading and supports, and density filtering. The PTOs excerpt sets up stress analysis and initializes density to 0.5, while PTOc initializes it to the volume limit.
Appendix A – PTOs
- PTOs_mbb sets up finite-element stiffness assembly, loads and supports, stress analysis, and density filtering before initializing density to 0.5 and entering its iteration loop.The function excerpt then assembles the stiffness matrix and solves for free degrees of freedom.Source: Appendix A · p. 15
Appendix B – PTOc
- PTOc_mbb uses the corresponding finite-element, loading, support, and filtering setup, then initializes density to the volume limit before its iteration loop.The excerpt identifies the function inputs, including the volume limit and density-filter radius.Source: Appendix B · p. 17