Multi-Dimensional Optimization of Loudspeakers
September 30, 2020:
A professional article by Tommaso Nizzoli, Thomas Gmeiner, Alfred J. Svobodnik (Mvoid Group) und Anton Saratov (DATADVANCE), presented at the virtual NAFEMS20 DACH conference on October 14, 2020
The fundamental principle of virtual product development is to optimize the performance of products and systems in the early phase of the development process. Engineering is the art of realizing desired functions by technical solutions embedded in a system of interacting elements. Multi-dimensional target requirements come with trade-offs to be made and the quest for the best product boils down to finding the sweet spot between all the interacting influences.
In multi-dimensional optimization of loudspeakers, the performance of a loudspeaker is improved by combining a parametric multiphysics model and an optimization algorithm. The optimization strategy is varying the geometrical parameters of the loudspeaker consequently modifying the FEM model which solves for the objective functions. Feeding these results into an optimizer generates a surrogate model which is trained by comparing each result with the surrogate functions. For faster convergence, the optimizer controls the parameter variation. As a result, the non-linear parameters, describing the large-signal performance of the loudspeaker are significantly improved.
Figure 1: Cross-section sketch of an electro-dynamic transducer and equivalent circuit model
An ideal loudspeaker translates a voltage signal via the excursion of a membrane into sound pressure by a linear dependency. The membrane is driven by a voice coil moving in the air gap of a magnet system. In a real system, the electrical properties of the voice coil depend on its position relative to the magnet system where the magnetic flux density of the magnetic circuit is not constant over the excursion. In the equivalent circuit, representing the behavior of the transducer translated to the electrical domain, the components are not linear.
Transducer developers have many knobs to turn for getting closer to a linear behavior over a wider excursion. Some knobs, however, have a negative impact on other parameters, e.g. a longer voice coil decreases the sensitivity. The desire is to find the optimum of many objective functions by varying many input parameters. For each variation, a new FEM for the magnet system is necessary, requiring processing time.
Optimization studies with detailed simulation models, require specific methods and algorithms. The time for one simulation run can be significant, so the number of runs must be controlled to get the best result in a given time frame.
In the following we describe the process of multi-dimensional optimisation of loudspeakers by combining a parametric multiphysics model and an optimization algorithm. The software pSeven was integrated into the Mvoid methodology.
Process of Multi-Dimensional Optimization of Loudspeakers
The basis is a digital model which design parameters are varied by an optimization algorithm with multiple objective (or target) functions. Data is exchanged between model and controlling optimization algorithm automatically to run iterations towards an optimal state.
For this purpose, a multiphysics model with its different interacting subsystems is developed to simulate the behavior of the transducer. The subsystems are the motor structure (magnet and coil), the mechanical system with the membrane and suspension and the acoustic system with cabinet and surrounding air. The input variable is a voltage signal, the output is the sound pressure. An FEA model of the magnet system yields the magnetic flux in the air gap, showing the stray-field as well and the decline of the flux density outside the airgap.
The driving force of the coil can be calculated in a lumped parameter model with non-linear elements. The coil with the metal core represents a complex impedance that varies with position. An applied voltage results in current which again results in a Lorentz force. Vice versa, the moving coil induces a current, called the back-electro-magnetic-force, resulting in an electrical damping effect.
The mechanical properties of the membrane and suspension is modelled also in an FEA model. Other influences such as thermal or viscous are neglected in this study, though it is possible to include these effects as well.
A set of so-called Thiele-Small (TS) parameters describe the functional performance of a loudspeaker and are represented in the equivalent circuit. The magnet system is now parameterised to vary its geometry (Figure 2). Each variation yields a new set of TS parameters.
Figure 2: Parametrization of loudspeaker geometry
In this study, we used the optimization methods of the Surrogate-Based Optimization (SBO) concept . It is based on the idea of internal auxiliary approximation models, which are trained inside the optimizer and used to propose the optimal candidates to be evaluated with a simulation model. Such internal model is retrained at each iteration using newly obtained simulation results to provide the best estimation of optimal parameters at each step. Such approach reduces the number of simulation model evaluations.
SBO algorithms support multi-objective problem statements , which are important to reveal the trade-off between different configurations of the system with conflicting target functions. It also provides special means to handle implicit constraints, which may appear for certain combinations of input parameters and allows to efficiently reuse existing simulation data. SBO allows to specify the budget (number of simulation model evaluations) explicitly, which allows to make efficient use of the available time for an optimal solution.
Figure 2 shows the cross-section of the magnet system with rotational symmetry. It is an advanced model with certain components introduced to improve performance. Nine geometrical parameters can be varied resulting in non-linear functions for Bl, Le, L2 and R2.
The motor structure coupled to the mechanical system and the acoustic environment yields the non-linear behavior at large voltage signals producing large excursions (Figure 3 and 4).
Figure 3: Loudspeaker at large excursion producing high sound pressure with distortion
Figure 4: Sound pressure level and harmonics
The simulation model of the loudspeaker performance allows to explore various non-linear effects, including levels of distortion and compression in the reproduced signal by extracting Bl, Le, L2, R2, stiffness and other parameters as functions of coil position. In order to define the optimization targets, those curves were postprocessed to obtain the scalar measures of main features. For example, Bl curve was optimized for symmetry, flatness and higher Bl value at zero position; Le, L2 and R2 were optimized for symmetry.
Some of those features compete and can be considered either in multi-objective optimization problem with up to 6 independent target and Pareto-frontier of optimal solutions as a result or scalarized into single objective to speed up the study.
Since all nine geometrical parameters are changed independently, there are configuration of the inputs, which lead to infeasible (impossible geometry) or errors in the solver. Such configurations are handled by the optimizer as implicit constraints, allowing to continue optimization search and avoid regions of inconsistent geometries.
Automated simulation allows to perform various parametric studies on the model. In many cases, especially for global exploration tasks, it is useful to perform a design of experiments (DoE) prior to run the optimization. DoE allows to examine the model behavior the whole parameter space and estimate the impact of the parameters on the responses of the model.
Such sensitivity analysis may reveal the unimportant parameters which can be dropped from the consideration to speed up and simplify the optimization problem solution.
Figure 5 shows the normalized effect of the geometry parameters variation on the Le curve symmetry metrics.
Figure 5: Impact of geometry parameters on variance of the Le symmetry
Even if all parameters are affecting the responses behavior and the dimensionality of the problem cannot be reduced, DoE points can be easily used as an initial starting sample for global search due to the nature of SBO method and the overall computation budget won’t be exceeded.
Optimization results and required number of evaluations depends on the problem statement. For simple scalarized problem with all targets combined into one, the optimization was performed within 150 simulation runs thanks to SBO and global optimum was revealed.
Comparison of initial and optimal Le, L2 and R2 curves, all optimized for symmetry, is shown on Figure 6.
Figure 6: Inductance curves for initial and optimal designs
It’s easy to notice the change in the curves shape with almost ideal symmetry of the optimal result in the whole range of coil excursion, which will result in significant reduction of nonlinear loudspeaker distortion.
The nonlinear performance of the loudspeaker was significantly improved by the modified geometry. The result was achieved in a couple of hours where it takes days for experienced engineers to achieve a similar result, if possible at all. The design space was fully explored, and the optimum balance of the objective functions found.
The study impressively shows how product development departments can benefit from the multi-dimensional optimization of loudspeakers and demonstrates the potential of virtual product development: Product designs can be varied over a wider range to explore the complete design space. The product under development can be consciously positioned within the explored space, therefore better understood and explained to the customer. User tests can be performed with the virtual prototypes to confirm decisions or explore new properties.
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If you want to discover more about multi-dimensional optimization of loudspeakers, we recommend you to watch the replay of the webinar “Multi-dimensional Optimization of Professional Audio Systems Using Advanced Multiphysical Models” held by Dr. A. Svobodnik at the virtual ALTI-Expo. Link to replay
We also recommend DATADVANCE’s technical abstract about “Adaptive Surrogate-Based Multi-Criteria Optimization”.
DATADVANCE is an independent software developer that offers its customers’ software solutions and consulting services for advanced data analysis, predictive modeling and design optimization, pSeven is a product of DATADVANCE.
Picture source: Mvoid Technologies GmbH, DATADVANCE & pSeven logo: Datadvance LLC