Shape Optimization Techniques

Understanding Shape Optimization in Modern Engineering

Shape optimization represents one of the most powerful computational techniques available to engineers today, enabling the design of structures and components that maximise performance while minimising material usage, weight, and cost. For engineering firms operating in Atlantic Canada, where industries ranging from offshore energy to shipbuilding demand innovative solutions, mastering shape optimization techniques has become essential for maintaining competitive advantage and delivering superior engineering outcomes.

At its core, shape optimization involves systematically modifying the geometry of a structure to achieve specific performance objectives while satisfying design constraints. Unlike traditional trial-and-error approaches, modern shape optimization leverages advanced algorithms, finite element analysis (FEA), and computational power to explore vast design spaces and identify optimal configurations that human intuition alone could never discover.

The Maritime provinces present unique engineering challenges—from harsh coastal environments and extreme temperature variations to the specific demands of the fishing, aquaculture, and energy sectors. Shape optimization techniques allow engineers to address these challenges by creating designs that are inherently more efficient, durable, and cost-effective for our regional context.

Fundamental Approaches to Shape Optimization

Parametric Shape Optimization

Parametric shape optimization represents the most straightforward approach, where the geometry is defined by a set of design variables or parameters. These might include dimensions such as thickness, radius, length, or angle measurements. The optimization algorithm then systematically varies these parameters to find the combination that best satisfies the objective function—whether minimising stress, reducing weight, or maximising stiffness.

For example, when designing a pressure vessel for Nova Scotia's offshore industry, engineers might parameterise the wall thickness, head geometry, and reinforcement locations. The optimization process would then identify the optimal values for each parameter that minimise material usage while ensuring the vessel meets all safety requirements under operating pressures of 10-50 MPa typical of subsea applications.

Free-Form Shape Optimization

Free-form optimization offers greater flexibility by allowing the boundary of a structure to move freely, subject only to manufacturing and geometric constraints. This approach uses techniques such as:

• Basis vector methods: Representing shape changes as combinations of predefined deformation patterns

• Mesh morphing: Directly manipulating finite element mesh nodes to achieve smooth geometry changes

• Implicit surface methods: Using mathematical functions to define boundaries that can evolve continuously

• CAD-based approaches: Linking optimization directly to computer-aided design software for seamless integration with manufacturing workflows

Free-form methods typically achieve weight reductions of 15-40% compared to conventional designs, depending on the application and constraints involved. However, they require more sophisticated computational resources and careful consideration of manufacturing feasibility.

Topology Optimization

While technically distinct from pure shape optimization, topology optimization deserves mention as a closely related technique that has revolutionised structural design. Topology optimization determines not just the shape but the optimal distribution of material within a design domain, often producing organic, lattice-like structures that were previously impossible to manufacture before the advent of additive manufacturing technologies.

Mathematical Foundations and Algorithms

Successful shape optimization requires a solid understanding of the underlying mathematical principles and computational algorithms. The optimization problem is typically formulated as:

Minimise: Objective function f(x) – such as weight, compliance, or stress

Subject to: Constraint functions g(x) ≤ 0 – such as stress limits, displacement bounds, or frequency requirements

Where: x represents the vector of design variables defining the shape

Gradient-Based Methods

Gradient-based algorithms remain the workhorse of industrial shape optimization due to their computational efficiency. These methods use sensitivity information—derivatives of the objective and constraint functions with respect to design variables—to guide the search direction. Common algorithms include:

• Sequential Quadratic Programming (SQP): Highly effective for problems with nonlinear constraints, typically converging within 20-50 iterations for well-posed problems

• Method of Moving Asymptotes (MMA): Particularly suited for structural optimization, handling large numbers of constraints efficiently

• Conjugate Gradient Methods: Effective for unconstrained or simply constrained problems with smooth objective functions

• Interior Point Methods: Excellent for large-scale problems with many inequality constraints

Sensitivity Analysis

Computing accurate sensitivities is crucial for gradient-based optimization. The two primary approaches are:

Discrete sensitivity analysis computes derivatives of the discretised (finite element) equations, offering high accuracy but requiring access to element-level information. This approach is well-suited for commercial FEA software integration.

Continuum sensitivity analysis derives gradients from the continuous governing equations before discretisation, providing mathematical elegance and mesh-independent results. This method is particularly valuable for free-form shape optimization where mesh quality must be maintained throughout the process.

Gradient-Free Methods

For problems with noisy objectives, discrete variables, or multiple local optima, gradient-free methods offer robust alternatives:

• Genetic Algorithms: Inspired by biological evolution, these methods explore design spaces through selection, crossover, and mutation operations

• Particle Swarm Optimization: Simulates social behaviour of bird flocks or fish schools to search for optimal solutions

• Simulated Annealing: Mimics the metallurgical annealing process to escape local optima

• Response Surface Methods: Build surrogate models to approximate expensive simulations, enabling efficient global optimization

While gradient-free methods require significantly more function evaluations—often 1,000 to 100,000 compared to 50-200 for gradient-based approaches—they offer greater robustness for challenging multi-modal problems common in real-world engineering applications.

Practical Applications in Maritime Industries

Marine and Offshore Structures

The shipbuilding and offshore energy sectors in Atlantic Canada benefit enormously from shape optimization techniques. Hull form optimization can reduce hydrodynamic resistance by 5-15%, translating directly to fuel savings and reduced emissions over a vessel's operational lifetime. For a typical fishing vessel operating out of ports along the Nova Scotia coast, even a 10% reduction in fuel consumption represents significant annual savings and environmental benefits.

Offshore platform components, including jacket structures, risers, and subsea manifolds, present excellent opportunities for shape optimization. These structures must withstand extreme loading conditions—wave heights reaching 15-20 metres during Atlantic storms, combined with current, wind, and operational loads. Optimised designs can achieve the required safety factors while reducing steel tonnage by 20-30%, with corresponding reductions in fabrication and installation costs.

Wind Energy Components

Nova Scotia's commitment to renewable energy development has created demand for optimised wind turbine components. Tower structures, foundation designs, and blade root connections all benefit from shape optimization. A well-optimised turbine tower can reduce steel requirements by 15-25% while maintaining or improving fatigue life under the cyclic loading conditions characteristic of Maritime wind regimes.

Foundation designs for onshore and potential offshore wind installations require careful optimization to account for Nova Scotia's varied geological conditions, from bedrock formations to glacial till deposits. Shape-optimised gravity-based foundations can reduce concrete volumes by 20-35% while ensuring adequate stability against overturning under extreme wind events.

Industrial Equipment and Process Systems

Manufacturing and processing facilities throughout the Maritimes rely on pressure vessels, heat exchangers, storage tanks, and piping systems that benefit from shape optimization. Optimised heat exchanger designs can improve thermal efficiency by 10-20% through better flow distribution and enhanced heat transfer surface geometries. This translates to reduced energy consumption and operating costs for industries ranging from food processing to pulp and paper manufacturing.

Software Tools and Implementation Strategies

Implementing shape optimization requires appropriate software tools and a systematic workflow. Leading commercial platforms include:

• ANSYS Mechanical with Design Optimization: Comprehensive FEA platform with integrated parametric and topology optimization capabilities

• Altair OptiStruct: Industry-leading topology and shape optimization solver with excellent manufacturing constraint handling

• SIMULIA Tosca: Powerful optimization suite integrated with Abaqus FEA for complex nonlinear problems

• modeFRONTIER: Multi-objective optimization platform supporting integration with virtually any CAE tool

• MATLAB with Optimization Toolbox: Flexible environment for custom algorithm development and research applications

Best Practices for Implementation

Successful shape optimization projects follow established best practices:

• Problem formulation: Invest adequate time defining objectives, constraints, and design variables. A well-formulated problem is half-solved

• Mesh quality: Ensure finite element meshes are sufficiently refined and maintain quality throughout shape changes. Adaptive meshing strategies are often essential

• Constraint handling: Include all relevant physical constraints, including manufacturing limitations, assembly requirements, and maintenance access

• Validation: Always validate optimised designs through independent analysis and, where possible, physical testing

• Documentation: Maintain detailed records of optimization parameters, convergence history, and design rationale for regulatory compliance and future reference

Emerging Trends and Future Developments

The field of shape optimization continues to evolve rapidly, driven by advances in computational power, manufacturing technology, and algorithmic development. Several trends are particularly relevant for engineering practice in Atlantic Canada:

Integration with Additive Manufacturing

Additive manufacturing technologies have removed many traditional constraints on geometric complexity, enabling the production of optimised designs that were previously impossible to fabricate. This synergy between optimization and 3D printing is particularly valuable for low-volume, high-value components common in offshore and marine applications.

Multi-Physics Optimization

Modern engineering challenges increasingly require simultaneous optimization across multiple physical domains—structural, thermal, fluid, and electromagnetic. Advanced multi-physics optimization frameworks enable holistic design approaches that account for complex interactions between different phenomena.

Machine Learning Integration

Artificial intelligence and machine learning techniques are being integrated with traditional optimization methods to accelerate convergence, handle high-dimensional design spaces, and learn from historical optimization data. Surrogate models based on neural networks can reduce computational costs by orders of magnitude while maintaining acceptable accuracy.

Uncertainty Quantification

Robust optimization methods that account for uncertainties in material properties, loading conditions, and manufacturing tolerances are becoming increasingly important. These approaches ensure that optimised designs perform well not just under nominal conditions but across the full range of realistic operating scenarios—essential for safety-critical applications in harsh Maritime environments.

Partner with Sangster Engineering Ltd. for Your Optimization Needs

Shape optimization techniques represent a powerful tool for achieving superior engineering designs that reduce costs, improve performance, and enhance sustainability. However, successful implementation requires deep expertise in both the underlying physics and the computational methods involved.

Sangster Engineering Ltd. brings decades of experience serving clients throughout Nova Scotia and Atlantic Canada, combining practical engineering knowledge with advanced analytical capabilities. Our team understands the unique challenges facing Maritime industries—from the demanding offshore environment to the specific requirements of regional manufacturing and infrastructure projects.

Whether you're seeking to optimise a new design, improve an existing product, or explore the potential of advanced manufacturing techniques, we're here to help. Contact Sangster Engineering Ltd. today to discuss how shape optimization can benefit your next project. Our Amherst-based team is ready to deliver innovative, cost-effective solutions tailored to your specific requirements.

Partner with Sangster Engineering

At Sangster Engineering Ltd. in Amherst, Nova Scotia, we bring decades of engineering experience to every project. Serving clients across Atlantic Canada and beyond.

Contact us today to discuss your engineering needs.