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We introduce a new computational tool called the Boundary Learning Optimization Tool (BLOT) that identifies the boundaries of the performance capabilities achieved by general flexure system topologies if their geometric parameters are allowed to vary from their smallest allowable feature sizes to their largest geometrically compatible feature sizes for given constituent materials. The boundaries generated by the BLOT fully define the design spaces of flexure systems and allow designers to visually identify which geometric versions of their synthesized topologies best achieve desired combinations of performance capabilities. The BLOT was created as a complementary tool to the freedom and constraint topologies (FACT) synthesis approach in that the BLOT is intended to optimize the geometry of the flexure topologies synthesized using the FACT approach. The BLOT trains artificial neural networks to create models of parameterized flexure topologies using numerically generated performance solutions from different design instantiations of those topologies. These models are then used by an optimization algorithm to plot the desired topology’s performance boundary. The model-training and boundary-plotting processes iterate using additional numerically generated solutions from each updated boundary generated until the final boundary is guaranteed to be accurate within any average error set by the user. A FACT-synthesized flexure topology is optimized using the BLOT as a simple case study.

The freedom and constraint topologies (FACT) synthesis approach [

Although the BLOT was originally created to plot the performance capability regions of FACT-designed architectured materials [

Desired DOFs (a); freedom and constraint spaces (b); synthesized topology (c); validated DOFs (d) and (e); topology’s geometric parameters (f); boundary of the topology’s achievable ranges of motion and natural frequencies corresponding to the translational DOF generated by the BLOT (g).

Suppose that the designer wished then to identify what geometric instantiation of the FACT-synthesized topology in Figure ^{3}, and a yield strength of 105 MPa. Once this boundary is known, the designer can select optimal design instantiations that simultaneously achieve the largest values of each of the desired capabilities from along the top-right portion of the boundary as shown in Figure

Note that although the BLOT introduced in this paper utilizes numerically generated data taken from a variety of corresponding design instantiations to train neural networks for creating an accurate model of FACT-synthesized topologies, the BLOT is also capable of using closed-form analytical models of the parameterized topologies if such models are available. Closed-form analytical models are, however, difficult to construct and their assumptions usually only produce valid results for specific topologies or for limited ranges of geometric parameters that define other topologies. Thus, this paper focuses on the theory necessary to train neural networks for creating models based on the results of numerically generated data because this approach is more general than closed-form analytical approaches. Note also that although the BLOT is introduced in this paper as a tool for optimizing the geometry of flexure system topologies, it could also be applied to a host of other diverse applications.

Prior to this paper, artificial neural networks have been used extensively to model and control a variety of mechanical systems including compliant mechanisms, robots, manipulators, and rigid linkages. Cheng and Patel [

Although few researchers have directly attempted system-performance boundary identification, its goal is similar to the goal of multiobjective optimization, which has been studied extensively. A multiobjective optimization problem (MOOP) deals with more than one objective function and aims at finding a set of solutions that optimizes all the objective functions simultaneously. Several methods have been proposed to solve the local or global Pareto-optimal solution set [

By combining the utility of the BLOT with the current FACT approach, a new advantageous approach emerges, which is unique from other existing design-optimization approaches. Whereas other approaches (e.g., topology optimization [

The specific contributions of this paper include the following: (i) a new fully automated tool is created and demonstrated (i.e., BLOT) that identifies the rigorous performance capability boundaries achieved by flexure topologies generated via FACT; (ii) a new algorithm is proposed within this tool for optimizing and iteratively training the architectures of artificial neural networks using data generated from specially selected numerical simulations of design instantiations along previously generated boundaries to create accurate models of general flexure topologies; (iii) a new approach is also introduced within the BLOT that combines existing multiobjective optimization methods to use these models for iteratively refining the convex and concave portions of a general topology’s performance capability boundary with an accuracy determined by the user.

This section summarizes how the BLOT generates models of FACT-synthesized topologies by training artificial neural networks using numerical simulations performed on various geometric instantiations of these topologies. Artificial neural networks, like the kind shown in Figure

Neural network example with three inputs, one output, and two hidden layers of neurons.

Each layer within a neural network consists of interconnected computational nodes called neurons as labeled in Figure

Once a sufficient number of known input and target output values have been numerically calculated, various methods can be employed to use these values to iteratively tune the weights,

The specific iterative process used in the BLOT to generate accurate models of general FACT-synthesized topologies is as follows. A user provides a MATLAB script with the independent geometric parameters that define the desired flexure topology. The smallest and largest values for each of these parameters are also provided to the script to specify the full design space over which the resulting model will accurately predict. The script then begins by generating all the possible design instantiations that result from applying every combination of these smallest and largest values with three other evenly spaced values in between them (e.g.,

The numerically generated data is then randomly divided among two different sets. 80% of the data is assigned to a training set and the remaining 20% is assigned to a testing set. The training set of data is used to train a neural network that possesses an initial-guess architecture with two hidden layers and an output layer with only one neuron. The number of neurons in the first hidden layer is initially set to 20 neurons and the number of neurons in the second hidden layer is initially set to zero. The modified mean squared error [

After training has stopped, the mean absolute percentage error [

Thus, the

Thus, using minimal computation, an initial model of general FACT-synthesized flexure system topologies can be identified. Note that many of the hard numbers chosen in this section were selected for rapidly generating accurate models of flexure system topologies with less than ~10 independent geometric parameters. The numbers in this section may need to be adjusted for scenarios with larger numbers of input parameters.

This section discusses the portion of the BLOT that utilizes the models of flexure system topologies generated from Section

The boundary-plotting algorithm first requires the smallest and largest values of each input parameter,

Progression of the SQP and then ALPS optimization algorithms for

The algorithm begins by supplying a randomly selected combination of permissible inputs, shown as the blue dot labeled

After the SQP algorithm terminates, the boundary-plotting algorithm of this paper continues the optimization process by supplying the combination of permissible inputs that map to the combination of outputs that produce the largest objective function value identified by the SQP algorithm to the Augmented Lagrangian Pattern Search (ALPS) [

Once the SQP and then ALPS algorithms have both run their full course for determining the maximum value of

The algorithm iterates this pattern of steps to identify the combinations of output values that lie farthest away along their prescribed directions defined by their corresponding

Once this process is complete, the MATLAB boundary function (The MathWorks Inc.) can be used to identify all the combinations of output values that lie along the boundary of a convex shape that circumscribes the output dots calculated and plotted using the system’s model during the iterative process. Many systems, however, produce a cloud of output dots that form a concave—not convex—region like the one shown in Figure

Generated cloud of output dots (a); convex boundary first identified (b); new objective function minimized to identify other output dots within circular or elliptical regions (c); elliptical contour diagram of the new objective function with an

A simple case study is performed to validate the performance of the boundary tracing algorithm of this section. Suppose that three normalized input parameters,

Output values of a full parameter sweep (a); boundary identified for the same example using BLOT (b).

This section provides the iterative process used to refine the initial neural network-system models generated using the theory of Section

This section uses the FACT-synthesized flexure system topology of Figure

Once the BLOT successfully created four accurate models of the flexure system topology of Figure ^{−6} for the plots of Figure

Boundaries of the topology’s achievable ranges of motion and natural frequencies corresponding to the translational (a) and rotational (b) DOF generated by the BLOT.

This paper introduces an optimization tool called the Boundary Learning Optimization Tool (BLOT), which utilizes an automated method for training neural networks to create accurate models of general flexure system topologies synthesized using the FACT approach. The BLOT uses these models to plot the boundaries of regions containing the combinations of desired performance capabilities that are achievable by different design instantiations of the FACT-synthesized topologies. Both the neural network training and boundary identification portions of the BLOT are iterated using additional boundary data until the final boundary is guaranteed to be accurate within a value specified by the user (e.g., less than 10% average error). A flexure system is provided and optimized as a case study.

The BLOT will be used in future works to generate Ashby-like material plots but for different architectured material topologies instead of for natural materials to compare the achievable properties of such microarchitectured structures composed of the same material. These plots will enable engineers to identify trends among architectured materials that result from microstructure instead of composition.

A preliminary version of this paper’s theory was first presented at ASME’s IDETC as paper no. DETC2017-67465.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Ali Hatamizadeh and Yuanping Song are co-first authors as their contributions to this paper are equal.

This work was supported by the Air Force Office of Science Research under Award no. FA9550-15-1-0321. Program officer Byung “Les” Lee is gratefully acknowledged. This work used computational and storage services associated with the Hoffman2 Shared Cluster provided by UCLA Institute for Digital Research and Education’s Research Technology Group.