⚡ electrical engineering

CalTop: A CalculiX-Based Framework for Large-Scale Three-Dimensional Topology Optimization and Aeroelastic Tailoring

This paper introduces CalTop, a shared-memory framework built on CalculiX and an Interior Point Optimizer that enables large-scale 3D topology optimization and aeroelastic tailoring of the NASA Common Research Model wing, demonstrating effective parallel performance and improved convergence through adaptive barrier strategies.

Original authors: Prateek Ranjan

Published 2026-10-12

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Prateek Ranjan

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the heavy, rigid skeletons of airplanes are replaced by structures that grow and shift like living tissue, finding the most efficient paths to carry weight and resist wind. This is the promise of a field called topology optimization. Instead of starting with a solid block of metal and cutting away the unnecessary parts, engineers use computers to let the material itself decide where it should exist. The computer acts as a digital sculptor, pouring virtual material into a design space and then slowly draining it away from areas that don't need to be there, leaving behind a shape that is perfectly tuned to the forces it must withstand. For decades, this process has been limited to small, simple parts because the calculations required to figure out the best shape for a massive, complex object are so immense that they overwhelm even the most powerful supercomputers.

A researcher at the University of Illinois and Georgia Tech has now built a new digital tool that pushes these limits further than ever before. They created a framework capable of designing the internal structures of full-scale airplane wings, handling millions of tiny design choices simultaneously. By combining a powerful structural analysis engine with a sophisticated mathematical optimizer, they demonstrated that it is possible to evolve the internal layout of a wing while it is being twisted and pushed by the wind. Their work shows that when you let the computer figure out the best way to arrange the material, the result is not just a lighter wing, but a fundamentally different kind of structure that looks nothing like the traditional ribs and spars found in current aircraft.

The researcher, led by Prateek Ranjan, developed a software system they call CalTop. This system is designed to run on shared-memory computers, which are machines where many processors work together on a single task using a common pool of memory. The core challenge they tackled was the sheer scale of the problem. To design a wing that is both strong and light, the computer must divide the wing into millions of tiny pieces, or elements, and decide for each one whether it should be solid metal or empty space. In their most ambitious test, they worked with a design containing nearly five million of these elements. To solve the physics equations for such a massive model, the software had to perform calculations that would take a single processor decades to complete. By spreading the work across dozens of processor cores, they managed to speed up the most time-consuming part of the process by more than thirty times, making the impossible suddenly feasible.

A key innovation in their approach was how they handled the "filtering" of the design. In these simulations, if the computer is allowed to make decisions for every single tiny element independently, the result is often a chaotic mess of disconnected specks that cannot be built. To prevent this, the researcher used a mathematical smoothing technique that ensures the material forms continuous, solid shapes with a minimum thickness. They built a custom system to assemble this smoothing process in parallel, allowing the computer to handle the massive amount of data without getting bogged down. This allowed them to run simulations on unstructured grids, which are flexible mesh patterns that can fit the complex curves of a real airplane wing, rather than being forced into a rigid, boxy grid.

The researcher tested their new framework on several standard engineering problems before tackling a full aircraft wing. They started with a simple beam, then moved to a short cantilever, and finally an L-shaped bracket. In each case, they compared different mathematical strategies for guiding the computer toward the best solution. They found that the strategy the computer uses to decide how quickly to tighten its constraints makes a huge difference. Some strategies, which are more conservative and slow, often get stuck in a local optimum—a solution that is good but not the best possible one. Other, more adaptive strategies allowed the computer to explore the design space more aggressively, finding solutions that were significantly lighter and stronger. In one test, the adaptive strategies found designs that were nearly forty percent more efficient than the conservative approach, proving that the path the computer takes to find the answer is just as important as the answer itself.

The true test came when they applied the framework to the NASA Common Research Model, a standard wing shape used by researchers worldwide to test new aerodynamic ideas. This was not a simple structural test where the wind load was fixed. Instead, they coupled the structural design with a fluid dynamics solver, a program that calculates how air flows over the wing. This created a feedback loop: as the computer changed the internal structure of the wing, the wing would bend and twist differently in the wind, which in turn changed how the air pushed on it. The computer had to adjust the angle of the wing and the distribution of material at the same time to keep the plane flying at the correct lift.

The results were striking. The computer did not simply reinforce the existing ribs and spars of the wing. Instead, it evolved a completely new internal architecture. The final designs featured a strong, continuous beam running along the length of the wing, much like a spine, but also developed a network of thinner, rib-like structures and web-like supports that connected the top and bottom surfaces. These secondary structures acted like a skeleton, transferring the aerodynamic loads from the skin to the main beam. The most successful strategies produced wings that were significantly stiffer and more efficient than those found by the conservative methods. The adaptive strategies managed to reduce the structural compliance, a measure of how much the wing bends under load, by more than seventy percent compared to the conservative approach.

Perhaps the most important finding was that the computer did not just find one solution; it found several different, equally valid internal structures depending on the mathematical path it took. This suggests that there is no single "perfect" wing, but rather a landscape of many excellent designs, each with its own unique arrangement of material. The researcher showed that by choosing the right mathematical approach, they could guide the computer to discover these superior, non-intuitive layouts. The resulting wings looked less like traditional metal boxes and more like the intricate, load-bearing bones of a bird's wing, with material concentrated exactly where the forces were highest and removed from everywhere else.

This work represents a significant step forward in the ability to design the next generation of aircraft. By proving that it is possible to run these massive, coupled simulations on shared-memory systems, the researcher has opened the door to designing wings that are not only lighter but also more aerodynamically efficient because their shape and structure are optimized together. The framework they built, CalTop, is now available to other researchers, allowing them to explore these complex design spaces without needing to build their own massive supercomputing infrastructure. While the current study focused on a specific wing shape under specific conditions, the methods developed here provide a foundation for tackling even more complex problems, such as wings that can change shape in flight or structures that must withstand extreme heat and stress. The era of designing aircraft by cutting away metal is giving way to an era where the computer grows the structure from the inside out, finding forms that human intuition alone could never imagine.

Technical Summary: CalTop – A CalculiX-Based Framework for Large-Scale 3D Topology Optimization and Aeroelastic Tailoring

Problem Statement

The design of next-generation aircraft, as outlined in roadmaps like Waypoint 2050, requires structural architectures that bridge aerodynamic, aeroelastic, and propulsion domains. Conventional airframe designs, restricted to standard arrangements of spars, ribs, and stiffeners, are insufficient for high-aspect-ratio wings and integrated propulsion systems where structural flexibility significantly redistributes aerodynamic loads. While topology optimization offers a systematic method to discover novel load-carrying architectures, its application to full-scale, high-resolution three-dimensional aero-structures remains computationally prohibitive.

The primary challenges identified are:

Computational Scalability: High-resolution discretizations (millions of design variables) required to resolve meaningful structural features (e.g., thin sheets, curved spars) lead to massive linear system solves and sensitivity calculations.
Numerical Scalability: The cost and convergence of underlying algorithms must remain tractable as discretization is refined. Standard sparse direct methods become memory-restrictive for 3D models.
Optimization Robustness: Interior-point methods (IPMs) must handle the highly non-convex nature of density-based topology optimization, where different update strategies can lead to distinct local optima.
Multidisciplinary Coupling: Full aeroelastic optimization requires simultaneous evolution of material distribution, structural deformation, aerodynamic loading, and trim conditions (angle of attack), complicating the design space.

Methodology

The paper introduces CalTop, a shared-memory framework for large-scale density-based topology optimization built upon the CalculiX finite-element solver and integrated with the Interior Point OPTimizer (IPOPT).

Software Architecture and Parallelism

Modular Design: The framework separates preprocessing, finite-element assembly, density filtering, sensitivity analysis, and optimization. It utilizes an Extended Design Structure Matrix (XDSM) to define data flow.
Shared-Memory Parallelism: Implemented using OpenMP and POSIX threads, the framework targets multi-core CPUs.
- CalFilt (Density Filter): A parallel assembly procedure generates sparse filter interactions. To avoid memory contention, threads write to thread-local binary shards which are merged post-assembly. The filter is applied as an out-of-core sparse matrix-vector product, streaming data from disk to handle millions of variables without constructing the global matrix in memory.
- CalTop (Linear Solve): Element assembly is parallelized, with each thread assembling contributions into thread-local sparse arrays before accumulation. The global system is solved using Intel MKL's PARDISO (a sparse direct solver).
Sensitivity Analysis: Analytical sensitivities are derived for compliance, volume fraction, center-of-gravity (CG), and aggregated stress functions. These are filtered using the transpose of the density filter operator.

Optimization Formulation

Solver: IPOPT is used to solve the nonlinear programming problem. Inequality constraints are handled via slack variables and logarithmic barrier functions.
Globalization: A KKT-error-based globalization strategy (Nocedal et al.) is employed, utilizing a non-monotone acceptance criterion to allow temporary increases in KKT error, facilitating more aggressive Newton steps in highly nonlinear problems.
Barrier Parameter Updates: The study evaluates four strategies:
1. Monotone: Conservative, fixed reduction.
2. LOQO: Predictor-corrector with non-monotonic updates.
3. Probing: Mehrotra's predictor-corrector method.
4. Quality Function: Explicitly balances primal feasibility, dual feasibility, and complementarity.
Scaling: Automatic gradient-based scaling is applied to objective and constraint functions to improve the conditioning of the KKT system.

Aeroelastic Coupling

For the aircraft-scale application, CalTop is coupled with the distributed-memory-capable SU2 flow solver via preCICE.

Architecture: A Multidisciplinary Design Feasible (MDF) approach is used.
Process: At each optimization iteration, IPOPT updates element densities and angle of attack ( $\alpha$ ). A staggered fixed-point iteration (MDA: Direct) solves for static aeroelastic equilibrium between SU2 (aerodynamics) and CalTop (structural).
Adjoint: Coupled adjoint systems compute sensitivities for the objective and constraints, accounting for the dependence of aerodynamic loads on structural deformation and topology.

Key Results

1. Computational Performance and Scaling

Strong Scaling: The density-filtering operation achieved a 31.21× speedup on 48 threads.
Cost Breakdown: Execution trace analysis of a single-thread design evaluation (NASA CRM model) revealed that linear-system operations (55.92%) and preprocessing (23.88%) dominate the computational cost. Matrix factorization is the primary bottleneck, motivating future distributed-memory parallelization.
Filtering: The out-of-core sparse filtering approach allows efficient execution on shared-memory architectures with millions of design variables.

2. Structural Benchmarks

Three benchmark problems were used to assess the influence of barrier parameter update strategies:

MBB Beam (Compliance Minimization): Adaptive strategies (LOQO, Probing, Quality Function) consistently achieved lower compliance (e.g., Quality Function: 64.37 J) compared to the Monotone strategy (89.07 J). All strategies satisfied first-order KKT conditions but converged to distinct local optima, highlighting the non-convex nature of the problem.
Short Cantilever (CG Constraints): The framework successfully handled explicit center-of-gravity constraints, redistributing mass from an infeasible initial state to satisfy bounds. Different strategies yielded distinct topologies and compliance values, with computational costs varying based on the optimization trajectory.
L-Beam (Stress Constraints): Introducing aggregated p-norm stress constraints significantly altered the optimization landscape. While adaptive strategies generally outperformed Monotone in compliance-only cases, the addition of stress constraints reduced the performance gap between strategies, though they still converged to different topologies. The Quality Function strategy was the most computationally efficient for the compliance-only case but became the most expensive when stress constraints were added.

3. Aeroelastic Application (uCRM-9 Wing)

The framework was applied to the topology optimization of the undeformed NASA Common Research Model (uCRM-9) wing at a jig state ( $C_L \approx 0.5$ , Mach 0.8).

Scale: The structural domain contained ~4.75 million elements (3.66 million active design variables).
Convergence: The Monotone strategy stalled at a high compliance (273.10 kJ) and failed to develop a clear load path. In contrast, adaptive strategies (Probing, Quality Function) reduced compliance to 74.77 kJ and 77.00 kJ, respectively, and achieved lower dual residuals.
Topology: The adaptive solutions developed physically interpretable architectures featuring:
- A dominant spanwise spar-like load path.
- Secondary rib- and web-like structures connecting the spar to the passive skins.
- Cylindrical web members providing through-thickness connectivity.
Coupling: The coupled formulation successfully evolved the material distribution, structural deformation, aerodynamic loading, and angle of attack simultaneously while satisfying lift and CG constraints.

Significance and Claims

The paper claims that CalTop provides a computational foundation for progressing from large-scale constrained structural topology optimization to aircraft-scale coupled aeroelastic design.

Scalability: By leveraging shared-memory parallelism and out-of-core filtering, the framework handles design spaces with millions of variables, a prerequisite for resolving fine structural features in aircraft wings.
Optimization Strategy: The study demonstrates that the choice of barrier parameter update strategy is critical. Adaptive strategies (LOQO, Probing, Quality Function) not only improve convergence rates but also guide the optimizer toward superior local optima in highly non-convex design spaces, a finding that holds true even in complex multidisciplinary problems.
Rigorous Assessment: The work emphasizes the importance of reporting stationarity (KKT residuals) and feasibility metrics alongside objective values, moving beyond visual inspection or objective stabilization to quantitatively assess convergence.
Multidisciplinary Integration: The successful coupling of a shared-memory structural solver (CalTop) with a distributed-memory flow solver (SU2) validates a pathway for high-fidelity, multidisciplinary topology optimization where aerodynamic loads and structural stiffness evolve concurrently.

The author notes that while the current implementation is limited to shared-memory parallelism, the execution traces identify linear system solves as the primary target for future distributed-memory parallelization to further scale the framework to even larger problems. Future work will also address projection-based filtering for sharper boundaries, nonlinear elasticity, and additional constraints such as buckling and frequency.