**Introduction:**

Adaptive structures can adjust and respond to changing conditions, and are revolutionizing various industries, including automotive, aerospace, civil engineering, medical, and many more. *“The adaptive structure reacts automatically and continuously to environmental influences and is fed solely by the “intelligent” behavior of the used starting materials. Materials Experience 2, 2021.”*

These structures offer improved performance, sustainability, and resilience. Simulations play a crucial role in optimizing their design, reducing weights, and maximizing their potential in diverse applications. This article examines the pivotal role of simulation, such as topology optimization, in advancing engineering adaptive structures in their real-world use cases for benefiting various industries.

**Understanding Adaptive Structures:**

Adaptive structures possess the unique capability to adapt and respond to varying external conditions. In the automotive industry, they include systems that adjust aerodynamic features, suspension, and engine behavior in response to driving conditions. In aerospace, morphing wings and shape-changing airfoils are examples of adaptive structures. In civil engineering, structures equipped with sensors and actuators that respond to seismic forces are considered adaptive. Additionally, adaptive structures find applications in medical devices, such as implants or prosthetics that adjust according to the body's needs. Designing and optimizing these structures requires addressing complex challenges related to their behavior under different loads and environments.

Designing strength into such engineering structures requires a preemptive understanding, making advanced simulation tools imperative to such efficient designs. Firstly, the design fundamental of such structures entails proper balance of stiffness that prevents excessive movement and deformations. These are calculated based on the worst case (real-world) scenarios, to make it reliable, and safe for real world use cases.

To assess such impacts on the design structures, engineers use simulation, factually, more accurate simulations give more accurate predictions to such scenarios. Current simulations are largely based on classical algorithms that model such physical phenomena and assess impacts on the mathematical replica of those structures.

The accuracy and efficiency of these numerical simulations rely on computational power and modeling approaches. Even with the most advanced computers it would take a very long time to get very high accuracy outcomes which allows engineers to reduce number of constraints and objectives from the problem. These constraints represent external conditions that would occur in the real-world, making these structures vulnerable to extreme outcomes during such events. Arguably, computational limitations remain a challenge for engineers, and definitely for these adaptive structures, resulting in partial or complete collapse. It indicates the important role of simulation that can deliver benefits with existing computational technology, promise an alternative and better approach.

**Role of Simulation in Adaptive Structure Design:**

Simulation techniques play a vital role in designing and evaluating adaptive structures across various industries. Finite element analysis (FEA) and computational modeling enable engineers to simulate the behavior of adaptive structures accurately.

Through simulation, engineers can analyze how these structures respond to different scenarios, predict their movement, deformation, and performance, and identify potential issues. Simulation allows engineers to make informed design decisions by optimizing shape, material usage, and actuator placement for enhanced performance.

The accuracy of this tool relies on the convergence of the solution as close as possible to the global optima, resulting in a better design that can tolerate multiple extreme conditions. Secondly, many variables are required to be included to achieve accurate modeling by incorporating many constraints and objectives to mimic real-world scenarios in detail. Thirdly, a simulation approach that can search for a large solution space containing many solutions is needed to get more optimized design structures to reduce the number of iterations.

Based on these features, Quantum information processing used in Quantum Algorithms can play significant role in advancing simulation used in engineering, opening scope efficiency by reducing convergence time and number of iterations during the mathematical calculations. It can efficiently replicate multiple conditions and constraints, as a quantum state can represent multiple states simultaneously, caused by two quantum phenomena superposition and entanglement. These are making quantum algorithms suitable for optimization and simulation tasks.

**Design Optimization through Simulation:**

Simulation-driven design optimization is critical for harnessing the full potential of adaptive structures across industries. Engineers can use simulation tools to evaluate different design iterations and assess structural behavior under various load cases. Simulation-guided actuator placement and decision-making processes optimize the performance of adaptive components, achieving the desired response with minimal energy consumption. Furthermore, simulation helps identify potential failure modes, improving overall reliability and safety.

For example, to design an adaptive structure in a building construction, different adaptive structures by calculating two values for each variant.

To ensure comparabilities, the outer dimensions, loading scenarios, and structural constraints remain constant across all variants.

The first value evaluates the potential mass-saving of the adaptive structure in comparison to a passive structure with the same topology but no actuation.

The second value represents the total mass of the adaptive structure.

**Structural Model:**

Firstly, for a given topology, it compares the minimum-weight designs of a structure without actuation to the designs of the same structure with actuators, solving two non-linear optimization problems. Secondly, for a given outer dimension, it evaluates different topologies with given outer dimensions and specific applications to provide guidelines. This comparison utilizes insights from structural mechanics and examines the influence of structural properties and the degree of static indeterminacy.

Structures composed of truss and beam elements, utilizing centric linear actuators such as hydraulic cylinders located in the center of the elements. To simplify and expedite computations, small deformations, and linear elastic isotropic material behavior are assumed. Instead, dynamic effects can be excluded and consider a linear elastic material model for each element with properties such as Young's modulus, tensile strength, mass density, and Poisson's ratio.

The equation of motion is expressed as KD = F, where K represents the system stiffness matrix, D is the solution vector, and F is the load vector comprising external forces and actuation inputs. The stiffness matrix is constructed from element stiffness matrices, which depend on the cross-sections of the elements. The system stiffness matrix and load vector are functions of design variables, including the cross-sectional areas and moments of inertia. The solution provides the structural responses for all load cases, which are utilized for further analysis [1].

**Structural Optimization: **

The computation of the minimum mass of the structure involves an optimization procedure. The objective function is the total mass, and there are non-linear constraints related to displacements and stresses. The stresses are evaluated at specific positions of each element, including the ends and the upper and lower edges of the cross-section. Only nodal loads are considered, and distributed loads are excluded to simplify stress assessment along the beam span. The buckling of compression elements is also checked.

The critical buckling force is calculated based on the cross-sectional properties, and the normal force must not exceed this value. To streamline the calculations and minimize the number of design variables, assumptions are made about the cross-sections. The feasibility of the structure, in terms of maximum allowable displacements, can be assessed at predetermined degrees of freedom, where the displacements must not exceed predefined limits [1].

The computation of the minimum possible mass of the structure involves an optimization procedure using the total mass as the objective function and several non-linear constraints for displacements and stresses. Since the analysis is limited to plane examples, the feasibility of the stresses is evaluated at four positions of each element: at either end of the element and the upper and lower edges of the actual cross-section.

To maintain simplicity, only nodal loads are considered, and distributed loads are not permitted, eliminating the need to check the stresses along the beam span. Additionally, each compressive element is checked for buckling. This involves computing Euler’s critical buckling force **N****b,e** for the element’s actual cross-section and ensuring that the absolute value of its normal force **N****e** does not exceed this critical buckling force. Assumptions regarding the cross-sections are necessary to streamline the calculations and minimize the number of design variables. Specifically, in this study, a square hollow section (SHS) is chosen, which can be described by only two independent variables: the cross-sectional area **A **and the moment of inertia **I**. Feasibility for maximum allowable displacements is evaluated at predefined degrees of freedom. The maximum displacement at those chosen degrees of freedom, denoted as **D****c**, must not exceed the predefined limit [1].

**Problem formulation (Truss design inspired by high-rise buildings/bridges):**

Mass minimization problem for passive structure reads:

Fig: Truss design, investigated variants with dimensions and actuator positions

Source:[1]

**Cantilever beam design problem:**

Fig. Topology optimization of the cantilever beam: (a) Initial design of the cantilever beam, (b) final design obtained, final designs obtained by the conventional method is (c) not revised or (d) directly taken.

Source: [2]

A short cantilever beam is used to illustrate the sensitivity property, with the design domain divided into 80 × 50 cells. To simplify the implementation of the topology optimization, ΦTVM(x) could be constructed parametrically by B-splines, RBFs and CS-RBFs, etc [2].

For example, the CS-RBFs;

Each support of the CS-RBF in the optimization has a radius of 0.3 m, centered at the cell vertex. Circular holes are initially imposed within the design domain, and the topology optimization procedure aims to minimize compliance with a volume constraint. The resulting designs, shown in Fig. (b–d), demonstrate the effectiveness of using **Φ****INT**** **to describe the structural domain, as it accurately confines the movements within the design domain and yields the best design result with the lowest compliance value. Conversely, using** Φ****TVM**** **leads to boundary violations and unsatisfactory designs if the sensitivity is not correctly adjusted. Fig. (d) displays the design result after manually setting the sensitivity along the boundary, although this operation becomes challenging for irregular and changing design domain shapes [2].

Therefore, accurate modeling and algorithms have the potential to overcome these challenges. A team of experts at BosonQ Psi has identified and overcome these design optimization challenges using an alternative quantum algorithm that has provided edges over many existing methods.

**Case Studies and Success Story: **

**BQP’s case study:**

Engineers at BosonQ Psi have formulated the mass optimization of a cantilever beam problem as follows:

Here, w(ρ) is the weight of the structure, F is the global load vector, K is the global stiffness matrix, and x is the global displacement vector (unknown). In the material density field ρ, the above equation can be related to the design variable points. The reserve factor (RF) is defined as the ratio between the given maximum compliance value to the compliance of the optimal structure.

The compliance can be calculated as:

Additionally, the binary design variable ρ presents the density of individual elements in the given structure with Solid elements represented as ρ(1), and the void element as ρ(0).

**Results:**

Topology optimization using Quantum algorithms designed by BosonQ Psi has achieved a similar optimal value with** only difference of 0.3%, however, converged in 1/4th of iterations showing promise for future engineering.** Faster convergence speed is achieved due to benefits of quantum information processing with existing HPC hardware making it feasible for industrial applications.

Numerous case studies illustrate the successful application of simulation in improving adaptive structures across industries. In the automotive sector, simulation-driven design has enhanced vehicle aerodynamics, resulting in improved fuel efficiency and performance. Aerospace engineers have utilized simulations to optimize morphing wings, enabling aircraft to adjust their shape in response to varying flight conditions, leading to increased efficiency. Civil engineering projects, such as adaptive bridges or buildings, have leveraged simulation to enhance structural resilience and responsiveness. Medical implant designs have benefited from simulations that optimize their adaptive behavior within the body.

**Future Directions and Emerging Trends:**

The integration of simulation technologies in adaptive structure design holds significant potential for the future across various industries. Advancements in real-time feedback control and quantum machine learning algorithms will further enhance engineers' ability to optimize adaptive structures. These advancements will facilitate the development of intelligent structures that continuously adjust their behavior based on real-time data from sensors, offering improved efficiency, responsiveness, and sustainability.

**Conclusion:**

Simulation serves as a cornerstone for enhancing the performance and sustainability of adaptive structures across industries, including automotive, aerospace, civil engineering, and medical. By accurately predicting and optimizing the behavior of these structures, simulation helps achieve improved efficiency, resilience, and safety. Looking toward the future, continued research and application of simulation-driven design strategies will unlock the full potential of adaptive structures across multiple industries.

**References:**

Geiger, Florian, et al. "A case study on design and optimization of adaptive civil structures."

*Frontiers in built environment*6 (2020): 94.

Cai, Shouyu, and Weihong Zhang. "Stress constrained topology optimization with free-form design domains."

*Computer Methods in Applied Mechanics and Engineering*289 (2015): 267-290.

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