Introduction
Topology optimization is a powerful technique that has gained significant attention in the engineering community. It allows designers to discover innovative and high-performance structural designs by optimizing the distribution of material within a given analysis domain. One of the primary objectives in topology optimization is the minimization of structural compliance, which has been a key focus of research in this field.
The finite-volume theory has emerged as a promising numerical approach for topology optimization, offering several advantages over traditional finite-element methods. This theory ensures the local satisfaction of equilibrium equations and the employment of compatibility conditions along edges in a surface-averaged sense, which helps mitigate numerical instabilities such as checkerboard patterns, mesh dependence, and local minima issues.
Despite the advantages of the finite-volume theory, the development of efficient computational tools for topology optimization has been a significant challenge. This contribution presents the TOP2DFVT, a Matlab-based platform that enables the optimization of 2D structures using the finite-volume theory for compliance minimization problems.
The TOP2DFVT algorithm incorporates both the SIMP (Solid Isotropic Material with Penalization) and RAMP (Rotational Approximation of Material Properties) material interpolation schemes, as well as sensitivity and density filtering techniques. These advancements result in a notably enhanced optimization tool, capable of generating checkerboard-free and mesh-independent optimized topologies.
Finite-Volume Theory for Topology Optimization
The finite-volume theory has been extensively studied for the analysis of linear elastic continuum structures, demonstrating its numerical efficiency and stability. The key features of this approach are the local satisfaction of equilibrium equations and the use of compatibility conditions along edges in a surface-averaged sense.
In the finite-volume theory, the analysis domain is discretized into rectangular subvolumes, where the displacement field within each subvolume is approximated using an incomplete quadratic Legendre polynomial expansion in the local coordinate system. The surface-averaged displacements and tractions are then employed to establish the local system of equations for each subvolume, as shown in Figures 1 and 2.
To improve computational efficiency and ensure the symmetry of the global stiffness matrix, a modified system of equations is introduced, relating the surface-averaged displacements and the resultant forces acting on the subvolume faces. This modification not only enhances the physical consistency of the formulation but also enables the use of optimized solvers for symmetric systems, reducing the overall computational cost.
Material Interpolation Schemes
The TOP2DFVT algorithm implements two widely used material interpolation schemes for topology optimization: the SIMP and RAMP methods.
The SIMP method employs a power-law function to penalize intermediate relative densities, as shown in the orange and yellow lines of Figure 3. This approach promotes a more distinct “black and white” design, where the optimized topologies tend to have well-defined material and void regions.
The RAMP method, on the other hand, utilizes a concave penalization function, as depicted by the green and blue lines in Figure 3. This approach results in a more gradual convergence towards the limit relative density values of 0 and 1, often producing optimized topologies with a smoother distribution of material.
The choice between the SIMP and RAMP methods depends on the specific requirements of the design problem, as each approach has its advantages. The TOP2DFVT algorithm allows users to select the material interpolation scheme that best suits their needs.
Topology Optimization Algorithm
The TOP2DFVT algorithm is designed to solve 2D compliance minimization problems using the finite-volume theory. The overall workflow of the algorithm is summarized in the flowchart shown in Figure 4.
The algorithm starts by initializing the design domain and material properties, followed by the discretization of the analysis domain into rectangular subvolumes. The finite-volume theory analysis is then performed to compute the surface-averaged displacements and tractions, which are used to establish the modified global system of equations.
The compliance minimization problem is formulated as a nested iterative loop, where the displacement field is computed by solving the modified global system of equations, and the design variables (relative densities) are updated using the Optimality Criteria (OC) method.
To address potential numerical instabilities, the TOP2DFVT algorithm incorporates both sensitivity and density filtering techniques. The filter radius is calculated based on the subvolume dimensions, ensuring that the optimized topologies are checkerboard-free and mesh-independent.
The algorithm also supports the implementation of the continued penalization scheme, where the penalty factor is gradually increased during the optimization process. This approach helps to promote a more distinct “black and white” design while ensuring the stability of the optimization process.
Numerical Examples
The performance and capabilities of the TOP2DFVT algorithm are demonstrated through several numerical examples, including a cantilever deep beam, a Messerschmitt-Bölkow-Blohm (MBB) beam, and an L-bracket beam.
Cantilever Deep Beam
The first example considers a cantilever deep beam subjected to a concentrated load, as shown in Figure 6. The optimized topologies obtained using the SIMP and RAMP material interpolation schemes, with and without filtering techniques, are presented in Figures 7 and 8, respectively.
The results demonstrate that the TOP2DFVT algorithm is capable of generating checkerboard-free and mesh-independent optimized topologies, particularly when the RAMP method is employed. The objective function histories for the SIMP and RAMP approaches are shown in Figure 9, highlighting the convergence characteristics of each material interpolation scheme.
Messerschmitt-Bölkow-Blohm (MBB) Beam
The second example involves the optimization of a half-MBB beam, as depicted in Figure 14. The optimized topologies obtained using the SIMP, top99neo, and RAMP approaches are presented in Figures 15, 16, and 17, respectively.
The results highlight the advantages of the finite-volume theory approach in terms of mesh independence, as the TOP2DFVT algorithm demonstrates a lower sensitivity to mesh refinement compared to the top99neo finite-element-based algorithm.
L-Bracket Beam
The final example considers an L-bracket beam subjected to a concentrated load, as shown in Figure 18. The optimized topologies obtained using the SIMP, top99neo, and RAMP approaches are presented in Figures 19, 20, and 21, respectively.
The results demonstrate the ability of the TOP2DFVT algorithm to handle stress concentration problems, generating optimized topologies that effectively distribute the load and minimize the compliance.
Conclusion
This study introduces the TOP2DFVT, a Matlab-based algorithm for the topology optimization of 2D elastic structures using the finite-volume theory. The algorithm incorporates the SIMP and RAMP material interpolation schemes, as well as sensitivity and density filtering techniques, to generate checkerboard-free and mesh-independent optimized topologies.
The numerical examples presented showcase the capabilities of the TOP2DFVT algorithm, highlighting its ability to handle compliance minimization problems and effectively address numerical instabilities associated with traditional topology optimization approaches. The algorithm’s performance is further benchmarked against the well-known top99neo finite-element-based algorithm, demonstrating the advantages of the finite-volume theory in terms of mesh independence and computational efficiency.
The TOP2DFVT algorithm represents a significant advancement in the field of topology optimization, providing researchers, educators, and practitioners with a robust and efficient tool for the design of high-performance structures. By making the source code publicly available, the authors hope to foster further advancements and collaborations in the field of structural optimization.
You can find the source code for the TOP2DFVT algorithm on GitHub, along with detailed instructions for its use. The authors welcome feedback and contributions from the community to continue improving and expanding the capabilities of this valuable optimization tool.
