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Using material distribution-based topology optimization, we optimize the bandpass design of a loudspeaker cabinet targeting low frequencies. The objective is to maximize the loudspeaker’s output power for a single frequency as well as a range of frequencies. To model the loudspeaker’s performance, we combine a linear electromechanical transducer model with a computationally efficient hybrid 2D–3D model for sound propagation. The adjoint variable approach computes the gradients of the objective function with respect to the design variables, and the Method of Moving Asymptotes (MMA) solves the topology optimization problem. To manage intermediate values of the material indicator function, a quadratic penalty is added to the objective function, and a non-linear filter is used to obtain a mesh independent design. By carefully selecting the target frequency range, we can guide the optimization algorithm to successfully generate a loudspeaker design with the required bandpass character. To the best of our knowledge, this study constitutes the first successful attempt to design the interior structure of a loudspeaker cabinet using topology optimization.

We use material-distribution-based topology optimization to design a three-port frequency dividing multiplexer at microwave frequencies. That is, by placing a good electric conductor inside the design domain, we aim to design a passive device that splits the incoming signal's frequencies into two frequency bands and transmits them to their respective output ports. The Helmholtz equation is used to model the time-harmonic wave propagation problem. We use the finite element method to solve the governing equation. The adjoint variable method provides the required gradients, and we solve the topology optimization problem using Svanberg's MMA algorithm. In this study, we present a technique for modeling the distribution of a good electric conductor within the design domain. In addition, we derive a power balance expression, which aids in formulating a series of three objective functions. In each successive objective function, we add more information and evaluate its impact on the results. The results show that by selecting a suitable objective function, we achieve more than 93.7 % transmission for both the frequency bands. Moreover, the numerical experiments suggest that the optimization problem is self penalized and is sensitive to the initial design.

This thesis employs material distribution-based topology optimization for wave propagation problems. In the material distribution approach, we define a material indicator function that models the presence and absence of material in a design domain. By placing material inside the design domain, the aim is to design a device that maximizes the output power or transmission of the system. The time-harmonic linear wave propagation problem is modeled using the Helmholtz equation. The governing equation is solved using the finite element method, and an artificial boundary condition is used to truncate the domain. Moreover, a gradient-based algorithm, the method of moving asymptotes by Svanberg, is used to solve the optimization problem. An adjoint method efficiently computes the gradients of the objective function with respect to design variables. 

This thesis considers two types of wave propagation problems: acoustic (Papers I-III) and electromagnetic wave propagation (Papers IV-V). In Papers I-II, we consider a bandpass design of a subwoofer. The aim of Paper I is to reduce the computational time required to evaluate the performance of a given subwoofer layout. To accomplish this, we develop a computationally efficient hybrid 2D-3D model. A full 3D model, as well as a lumped model, validate the hybrid model's results. Paper II focuses on optimizing the topology of a subwoofer using the computationally efficient hybrid model from Paper I for single as well multiple frequencies. In Paper III, we design a highly efficient uni-directional linear acoustic waveguide. Moreover, we also challenge the use of the term acoustic diode for such uni-directional linear acoustic waveguides in literature. Paper IV deals with the design of a microwave frequency dividing multiplexer, which splits the incoming signals into two frequency bands and delivers them to their respective output ports. In Paper V, we use the adjoint method to perform the sensitivity analysis of a coupled plasmonic problem where a Helmholtz equation is coupled to the Poisson equation. We validate the sensitivities computed using the adjoint method with the finite difference approach.

In material distribution-based topology optimization, we place material inside a design domain to extremize an objective function. The optimization problem is solved using a gradient-based algorithm. An efficient way to compute the gradients is to use the adjoint method. This study performs the sensitivity analysis of a coupled plasmonic problem using the adjoint method. More precisely, a TE-polarized Helmholtz equation is coupled to a Poisson equation. The sensitivity analysis of the coupled plasmonic problem poses some challenges stemming from the complex solution of the plasmonic problem. Therefore, we first consider a model problem whose structure is similar to the main problem in some ways but is simpler to study. After examining the model problem, we perform the sensitivity analysis of the coupled plasmonic problem, highlighting key differences between the two problems.

A subwoofer generates the lowest frequency range in loudspeaker systems. Subwoofers are used in audio systems for live concerts, movie theatres, home theatres, gaming consoles, cars, etc. During the last decades, numerical simulations have emerged as a cost- and time-efficient complement to traditional experiments in the design process of different products. The aim of this study is to reduce the computational time of simulating the average response for a given subwoofer design. To this end, we propose a hybrid 2D–3D model that reduces the computational time significantly compared to a full 3D model. The hybrid model describes the interaction between different subwoofer components as interacting modules whose acoustic properties can partly be pre-computed. This allows us to efficiently compute the performance of different subwoofer design layouts. The results of the hybrid model are validated against both a lumped element model and a full 3D model over a frequency band of interest. The hybrid model is found to be both accurate and computationally efficient.

By using topology optimization, we consider the problem of designing a passive acoustic device that allows for one-way flow of sound waves; such a device is often colloquially referred to as an acoustic diode. The Helmholtz equation is used to model the time harmonic linear wave propagation together with a Dirichlet-to-Neumann (DtN) type boundary condition, and the finite element method is used for discretization. The objective of this study is to maximize the wave propagation in one direction (from left to right) and minimize the wave propagation in the reverse direction (from right to left) for planar incoming waves. The method of moving asymptotes (MMA) solves the optimization problem, and a continuation approach is used for the penalizing intermediate design variables. The results for the optimized waveguide show that more than 99.8% of the power of planar incoming waves get transmitted from left to right while less than 0.3% gets transmitted in the reverse direction for planar incoming waves in the specified frequency range. Since a true diode is a non-reciprocal device and here we used a linear acoustic wave model, which is basically reciprocal, we discuss details about how it appears to be possible to obtain a one-way waveguiding effect using this linear model.

We use material distribution-based topology optimization to optimize the design of a bandpass subwoofer enclosure. The objective is to maximize the subwoofer's output power for a single frequency as well as for a range of frequencies. A linear electromechanical transducer model is combined with a hybrid 2D-3D model for sound propagation to model the subwoofer's performance. The adjoint variable approach is used to compute the gradients of the objective function with respect to the design variables, and the Method of Moving Asymptotes (MMA) is used to solve the topology optimization problem. To manage intermediate values of the material indicator function, a quadratic penalty is added to the objective function, and a non-linear filter is used to obtain a mesh independent design. By carefully selecting the target frequency range, we can guide the optimization algorithm to successfully generate a subwoofer design with the required bandpass character. This study constitutes, to the best of our knowledge, the first successful attempt to design the interior structure of a loudspeaker using topology optimization. The success is much due to the hybrid 2D-3D approach, which reduces the computational effort significantly with only small effects on the modeling accuracy.