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Microcontrollers, specialized sensors, circuit design software (Proteus/MATLAB), and standard simulation platforms for Balanced Laminate Design Ansys.

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Yes. Full simulation packages, source code/RTL, circuit schematics, university-format report, presentation slides, and viva Q&A are provided for Balanced Laminate Design Ansys.

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Balanced Laminate Design Ansys Project | Balanced Laminate Design Ansys Design and Implementation | Balanced Laminate Design Ansys IEEE Project

Exploring Feasible Design Spaces for Heterogeneous Constraints Amir M. Mirzendehdel, Morad Behandish, and Saigopal Nelaturi Palo Alto Research Center (PARC), 3333 Coyote Hill Road, Palo Alto, California 94304

Abstract

We demonstrate an approach of exploring design spaces to simultaneously satisfy kinematics- and physics-based re- quirements. We present a classiﬁcation of constraints and solvers to enable postponing optimization as far down the design workﬂow as possible. The solvers are organized into two broad classes of design space ‘pruning’ and ‘exploration’ by considering the types of constraints they can satisfy. We show that pointwise constraints deﬁne feasible design sub- spaces that can be represented and computed as ﬁrst-class entities by their maximal feasible elements. The design space is pruned upfront by intersecting maximal elements, without premature optimization. To solve for other constraints, we apply topology optimization (TO), starting from the pruned feasible space. The optimization is steered by a topological sensitivity ﬁeld (TSF) that measures the global changes in violation of constraints with respect to local topological punctures. The TSF for global objective functions is augmented with TSF for global constraints, and penalized/ﬁltered to incorporate local constraints, including set constraints converted to diﬀerentiable (in)equality constraints. We demon- strate application of the proposed workﬂow to nontrivial examples in design and manufacturing. Among other examples, we show how to explore pruned design spaces via TO to simultaneously satisfy physics-based constraints (e.g., minimize compliance and mass) as well as kinematics-based constraints (e.g., maximize accessibility for machining).

Figure: Model & System Architecture for Balanced Laminate Design Ansys

Feasible Design Space, Composing Solvers, Workﬂows, Maximal Elements, Sensitivity Fields

Ntroduction

Mechanical design problems require reasoning about di- verse, multiple, and often conﬂicting objectives and con- straints arising from requirements across a product’s life- cycle. The key engineering design challenge lies in travers- ing the trade space of these requirements to synthesize feasible designs. This challenge has recently been ampli- ﬁed by rapid advances in manufacturing processes. Light- weight, high-performance, and multi-material composite structures with complex geometry and material distribu- tion can now be fabricated using various additive manu- facturing (AM) processes.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

Yet, Existing Computer-Aided

design (CAD) systems are far behind in their represen- tations and algorithms to navigate the high-dimensional trade spaces that grow exponentially in the number of available decisions per spatial elements. Additional func- tional constraints such as manufacturability, ease of as- sembly, motion in presence of obstacles, and aesthetics dramatically increase the trade space complexity.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

Specialized domain-speciﬁc computational tools are used to generate designs that satisfy speciﬁc types of func- tional requirements. For example, to maximize a part’s performance with as little cost or material as possible, one

May Employ Topology Optimization (To) Tools . In

most TO approaches, an objective function is deﬁned over a design domain in terms of the physical performance (e.g., strength and/or stiﬀness) with additional constraints on total mass or volume as well as boundary conditions (e.g., loading and/or restraints) that often account for interfaces with other parts. TO produces valid designs with nonin- tuitive shapes, topologies, and material distributions that meet physical requirements, but is rarely aware of other important design criteria such as kinematic constraints.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

On the other hand, to ensure collision-free motion of a part in an assembly, one may need to examine its free conﬁgura- tion space to guarantee collision avoidance. Similarly, for subtractive manufacturing (SM), the machinability of a designed part is predicated on whether the volume to be removed from a raw stock is accessible within the cut- ting tool assembly’s non-colliding conﬁgurations [7, 8]. For AM, one may need to consider the part’s morphology, min-

Imum Feature Size, And Skeleton Hybrid Manufac-

turing (combined AM and SM) requires more complicated logical reasoning . These problems require nontrivial interference analysis of shapes in relative motion that rely on diﬀerent tools of reasoning than physics-driven design tools such as TO. The latter often ignore motion related constraints by considering them out-of-scope.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

1.1. Kinematic, Physical, & Manufacturing Constraints Generating practical designs requires simultaneous rea- soning about shape, motion, materials, physics, manufac- turing, and assembly, among other factors. For example, a machine part that moves relative to other parts in a mechanical device has to avoid collisions with both sta- tionary and moving obstacles .

Figure: Model & System Architecture for Balanced Laminate Design Ansys

These Requirements

are imposed as kinematic constraints, expressed in terms of pointset containment or non-interference (Section 3.3). The same part has to sustain mechanical loads at its joints

Arxiv:1907.01117V2 [Cs.Cg] 10 Jul 2019

or points of contact with other parts. These requirements are imposed as physical constraints, expressed in terms of (in)equalities of mathematical functions that represent physical response (e.g., bounds on deﬂection or stress).

Figure: Model & System Architecture for Balanced Laminate Design Ansys

Moreover, the part has to be manufacturable with one or more AM or SM capabilities . Manufacturability con- straints can be of both kinematic and physical types; for instance, accessibility in SM [7, 8] and post-processing of AM (e.g., support removal ) are of predominantly kine- matic nature, whereas achieving desired material proper- ties in AM requires in situ physical analysis . With few exceptions (e.g., TO for AM with minimized support ) TO algorithms are not developed with manufacturability provisions built into their objective functions.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

1.2. Generating Feasible Designs with Multiple Solvers Diﬀerent computational services, herein called solvers, are used to assist the design process with heterogeneous constraints by providing either analysis tools to evaluate the performance of one or more given designs, or syn- thesis tools to generate one or more designs that satisfy a given set of performance criteria analysis.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

We Distin-

guish between these two types of solvers as forward and in- verse problem solvers (‘FP/IP-solvers’), respectively (Sec- tion 3). Speciﬁcally, generative design tools are IP-solvers which solve the inverse problem by systematically gener- ating candidate designs and evaluating their performance (using FP-solvers) to guide reﬁnement of designs until the criteria are met.

Figure: Model & System Architecture for Balanced Laminate Design Ansys

It is unlikely that a single IP-solver is capable of si- multaneous reasoning about all design criteria (e.g., ob- jective functions and constraints).

Therefore, A Typical

computational design workﬂow requires carefully organiz- ing and reasoning about several multidisciplinary solvers to address heterogeneous design criteria. While every IP- solver reasoning about a subset of these criteria may pro- duce designs that are feasible (and ideally, optimal) only with respect to the speciﬁc criteria it considers, it provides no guarantees about the rest of the criteria. Diﬀerent IP- solvers are thus likely to generate designs distinct from one another, while none of them simultaneously satisﬁes all criteria. Except for extremely simple criteria, it appears impossible to combine these solutions in any obvious way that preserves the constraints satisﬁed separately by each solver, or at least provides the best compromise. Even if such a solution exists, there may not exist any particular ordering of these solvers to ﬁnd it, simply because each solver performs premature optimization with regard to the subset of criteria it cares about.

Consider the problem of designing a car hood latch (adopted from ). An initial design domain with bound- ary conditions is provided, and the goal is to ﬁnd a design that is as stiﬀas possible with the least mass.

Ore-

over, the latch has to be free to rotate clockwise by 20 degrees around a revolute joint, without exiting the initial design domain, so that it would not collide with other parts that are possibly outside that envelope. Functional load- bearing surfaces where the latch mates with other parts are shown in Fig. 1. All feasible designs must retain these surfaces as speciﬁed. The said requirements immediately suggest using two IP-solvers that are well-positioned to deal with them (each solver satisfying a subset of them): • Let Unsweep be a solver that generates a design that remains within a given region of space while mov- ing according to a prescribed motion (in this case, a clockwise rotation of 21 degrees).

• Let Pareto Be A Solver That, Starting From An

initial design, generates a design on the Pareto front of the two objectives (compliance and mass), i.e., one that satisﬁes the stiﬀness requirement and the given boundary conditions with minimal mass.

Figure 1: Functional requirements for designing a car hood latch, including (a) containment under given motion after assembly; and (b) boundary conditions on interfaces for assembly.

Using these solvers separately, one may generate two distinct designs, as illustrated in Fig. 2. However, there is no clear operation with which to combine these two, in order to generate a design that satisﬁes both kinematics- and physics-based constraints. For example, the intersec- tion of the two designs, shown in Fig. 2 generates a design that does not violate the constraints satisﬁed by Unsweep, because every subset of the unswept volume also satisﬁes the containment constraints. However, the constraints sat- isﬁed by PareTO are no longer satisﬁed, because the load paths are changed due to the changed topology and the compliance target is no longer met.

In this case, Unsweep has a property that its solution can be interpreted not as a single design, but as a representa- tion of all designs that satisfy the containment constraints.

This family of designs is closed under set intersection, i.e., intersecting any of the feasible designs with another set leads to another feasible design.

Similarly, Pareto Can

generate a family of designs that satisfy compliance re- quirements for diﬀerent mass budgets. However, this fam- ily is not closed under set intersection. The goal of this paper is to show how we can exploit such information to organize the design workﬂows such that they produce so- lutions that satisfy all criteria.

It is possible to obtain a feasible solution to the latch design problem by using the same set of IP-solvers, if the Figure 2: Two partially feasible designs are generated separately by two solvers.

Each solver optimizes for a subset of constraints. Arbitrarily combining them (e.g., via set intersection in this example) may result in an infeasible design with respect to all constraints.

workﬂow is organized diﬀerently, as shown in Fig. 3. Sup- pose we ﬁrst solve for the containment constraint using Unsweep and generate a valid design that does not exit a given envelope while moving. We can use this design as the initial design input to PareTO and optimize its shape and topology to achieve the compliance target with minimal mass.

This Approach Works Because Pareto Is A Mate-

rial reducing solver, i.e., its solutions remain strictly con- tained within the initial design. Hence any design gener- ated downstream will be faithful to the containment con- straint that was satisﬁed upstream. The same argument is not true if the order of applying the solvers is swapped.

There is no reason to believe that applying Unsweep to a topologically optimized solution of PareTO will remain on the Pareto front. The fundamental diﬀerences between the two IP-solvers should be taken into account when deciding on their arrangement in a workﬂow—in this case, choosing between parallel or sequential execution and the proper or- der for the latter. We elaborate on these diﬀerences in the classiﬁcation of solvers in Section 3.

Figure 3: Both constraints can be satisﬁed by ﬁrst ﬁnding a shape that satisﬁes the containment constraint and then using this shape as the input to TO (or any other material reducing IP-solver).

Figure 4 shows a one-parametric family of solutions on the Pareto front obtained when using mass and compli- ance as competing objectives.

Pareto Provides A Clear

advantage over classical TO with a single solution, by pro- ducing many alternatives some of which might satisfy ad- ditional constraints that were not accounted for in TO.

This generate-and-test approach is plausible, and in fact, is a common strategy for dealing with heterogeneous con- straints. However, it turns out that none of the Pareto- optimized designs in this case will not remain within the envelope after a clockwise rotation of 21 degrees.

In general, a good rule of thumb to organize solvers in a workﬂow is to call the ones that produce the broadest families of designs earlier. The upstream IP-solvers should generate a large number of designs, as opposed to ﬁxing one or few designs, to provide more ﬂexibility for down- stream solvers. The downstream solvers can be FP-solvers, testing for new constraints, or IP-solvers, applying further optimization. However, each solver may prematurely op- timize designs that may fail evaluation criteria considered in downstream solvers. The “blind” process of generating and testing designs without carefully considering proper- ties of the workﬂow – and the associated feasible design space for each solver in the workﬂow – will scale poorly with increasing number of constraints/solvers and their complexity. A systematic approach to arranging solvers into workﬂows that guarantees satisfying new constraints without violating the already satisﬁed constraints is much demanded, and is the subject of this paper.

Figure 4: None of the topologically optimized design alternatives obtained from applying PareTO to the initial design will satisfy the containment constraint. Generate-and-test does not always work.

The Main Contributions Of This Paper Are:

1. classiﬁcation of IP-solvers based on the properties of design constraints (e.g., locality and continuity) that

Each Solver Can Handle;

2. organization of computational design workﬂows with heterogeneous constraints into two stages of design space ‘pruning’ and ‘exploration’, where pruning pro-

Vides The Feasible Design Space For Exploration;

3. reconciling set constraints with inequality constraints and deriving conditions under which they can be ex- pressed in a strictly local (i.e., ‘pointwise’) fashion; 4. pruning design spaces, treated as ﬁrst-class entities,1 1In the lexicon of programming languages, a construct is said to have ﬁrst-class status if it can be passed as an argument, returned from a subroutine, or assigned into a variable .

by intersecting their representative maximal elements

Deﬁned Implicitly By Pointwise Constraints;

5. strategies to compose FP-solvers into a generative and iterative design process by combining kinematic, man-

Ufacturing, And Physical Constraints; And

6. illustrating real-world design problems solved using multidisciplinary solvers, which are rarely addressed in the (otherwise siloed) areas of related research.

In Section 2, we review some of the recent works in related areas of computational design and manufacturing. In Section 3, we present basic deﬁnitions (FP/IP-solvers, global vs. local constraints, and design/performance space terminology). We present a classiﬁcation of solvers based on the types of constraints they can handle, and propose a systematic approach for organizing them into workﬂows.

In Section 4, we show that an important class of con- straints can be satisﬁed upfront by pruning the design space in the early stages without premature optimization, allowing ﬂexibility for downstream design decisions. The feasible design subspaces are described uniquely and com- pletely by their maximal elements. These representative elements are, in turn, implicitly deﬁned by a point mem- bership classiﬁcation (PMC) test implemented in terms of strictly local (i.e., ‘pointwise’) constraints. We demon- strate examples of kinematics-based constraints that ap- pear in assembly, packaging, and manufacturing.

In Section 5 we deal with more general global and local constraints, including (in)equality constraints that specify physics-driven requirements or manufacturability.

We use a Pareto-tracing levelset topology optimization (PareTO) which uses ﬁxed-point iteration to satisfy multiple design criteria guided by augmented TSFs. We demonstrate that accessibility constraints can be incorpo- rated in TO by ﬁltering the augmented TSF using the overlap measure of cutting tool and optimized part.

We conclude in Section 6 by a discussion of the limita- tions of our approach and proposed future directions.

Related Work

Real-world design problems often involve solving multi- objective optimization problems where the goal is to ﬁnd the best trade-oﬀbetween multiple competing objective functions. Classical methods such as linear programming (e.g., the ‘simplex’ algorithm), nonlinear programming (e.g., the steepest descent and conjugate gradients), or Newton-Raphson are limited to single-objective op- tion, ﬁnding the global optimum is NP-hard . Numer- ous approaches have been developed to converge to locally optimal solutions in reasonable computation time to multi- objective problems across diﬀerent disciplines.

Unlike single-objective optimization, a total ordering for feasible solutions may not be possible in multi-objective optimization, i.e., there may not exist a single “best” solution due to competing objectives. However, feasible solutions may be partially ordered according to Pareto eﬃciency—also known as Pareto–Koopmans eﬃciency or dominance [24, 25]. Pareto-optimal solutions are locally optimal (according to Pareto-eﬃciency) where improving one objective comes at the expense of at least one other objective . The collection of all Pareto-optimal solu- tions is referred to as a Pareto front, which represents a curve, surface, or higher-dimensional manifold in the de- sign space for two, three, or higher number of competing objectives, respectively. Tracing a Pareto front is a key challenge in multi-objective and multi-disciplinary design optimization. We will not attempt an exhaustive review of all approaches such as gradient-free methods including rule-based techniques and evolutionary algorithms . Rather, we focus on a special class of algorithms that automatically generate designs on the Pareto front of multiple objectives given an initial design.

TO [3, 36, 37] has emerged as a practical class of com- putational methods for designing high-performance light- weight structures and has been applied in numerous areas such as designing automobiles components , aircraft components [39, 40], spacecraft modules , cast parts

Ucts. Numerous Approaches Such As Density-Based ,

levelset-based [47, 48], and evolutionary methods for TO have been developed.

Optimal Design For Manufacturing

TO typically focuses on optimizing designs for perfor- mance (e.g., physical response to loads during operation) but less on other important factors such as manufactura- bility. Apart from traditional processes such as machin- ing and molding, more recent technologies such as AM have introduced the ability to fabricate complex topologi- cally optimized designs while presenting new manufactur- ing challenges . Process limitations must be considered during the design/optimization stage as much as possible to avoid repeated prototyping and iterations until the op- timized designs are manufacturable. Speciﬁcally, applying corrections to the geometry or topology of a solu- tion after TO to make it manufacturable may sacriﬁce the achieved optimality.

One solution is to impose design rules obtained from do- main expertise and experience. These rules relate speciﬁc combinations of shape, materials, and process to impose simpliﬁed constraints that can be built into the TO frame- work to restrict the feasible design space. For example, when designing for AM via fused deposition modeling us- ing polymers, one should require that all facets oriented at an angle greater than 45 degrees with respect to the build direction be supported with additional scaﬀolding material. When designing for casting and injection mold- ing processes, one should ensure that the part has fea- tures of almost uniform thickness and no entrapped holes are present, so that the mold can be removed and the molten material cools down uniformly throughout the part [53, 54]. When designing for wire- or laser-cutting, one should ensure that the ﬁnal design has a uniform cross- section, i.e., is 2.5D along the cutting direction.

These

constraints can be imposed during TO through ﬁltering of the sensitivity ﬁeld as illustrated in Fig. 5. Figure 5: Examples of enforcing manufacturability constraints by TSF ﬁltering.

Diﬀerent ﬁlters (c, e, g) – plotted along the cross- sections shown on the right – produce diﬀerent solutions (d, f, h). Another important AM consideration is the manufac- turing resolution, which can be directly incorporated into the TO algorithm as a minimum feature size constraint through either local gradient constraints or TSF ﬁlter- ing (Fig. 6). It is also possible to reduce the amount of support structure needed in an AM by either ﬁnding a good build orientation or TSF ﬁltering. Build orienta- tion optimization often involves solving a multi-objective problem taking into account other factors such as surface quality [56, 57], build time , or manufacturing error . TSF ﬁltering, on the other hand, can be achieved by penalizing overhang surfaces [59, 60], penalizing undercut surfaces , or augmenting new TSFs .

Other AM constraints may not be as straightforward. For instance, optimizing designs with respect to shrink- age and warpage during material phase changes within the AM process may require solving a multi-physics prob- lem at every iteration [17, 63].

Haracterizing Material

Figure 6: Constraining minimum feature size via TSF ﬁltering . properties of AM parts is also challenging due to process- induced anisotropy. Lack of inter-layer adhesion and vary- ing thermal history at every point can introduce unin- tended porosities and consequently aﬀect material behav- ior.

Although there are initial results for considering anisotropy in TO under certain assumptions , solving the problem by simultaneously optimizing the geometry, topology, and process parameters involves costly in-situ manufacturing simulation or process planning [7, 13].

Esign For Motion-Related Constraints

In addition to the above examples of performance and manufacturing requirements, there are other important de- sign criteria that involve spatial reasoning about the inter- actions of moving (translating and rotating) shapes such as collision avoidance, packaging, robot motion planning, and accessibility analysis. These requirements cannot be easily enforced by design rules, TSF ﬁltering, or other techniques commonly used in TO. Rather, they are often expressed as set constraints, i.e., statements in the language of sets (e.g., in terms of aﬃne transformations, Boolean opera- tions, and containment) rather than the language of real- valued functions used for (in)equality constraints in TO.

A broad class of inverse problems in practical design and manufacturing reduce to solving set constraints formulated in the conﬁguration space of rigid motions [6, 65–67].

Although the problems with set constraints are common and of signiﬁcant importance, they are not mainstream in design/optimization workﬂows due to non-smoothness and computational intensity . There are instances of TO frameworks that deal with motion-related problems in an ad hoc manner; for instance, in modeling collision and contact when designing compliant mechanisms or parts made of hyperelastic materials that undergo large deformations part [70, 71].

However, It Is Not Immedi-

ately obvious how set constraints can be incorporated in a systematic fashion into the design/optimization process without incurring prohibitive computation costs of spatial analysis at every iteration. The next section presents a diﬀerent classiﬁcation of constraints that enables design space pruning and exploration, in which set constraints are also restated in terms of (in)equalities of functions.

Figure 7: Solving heterogeneous (e.g., kinematic, physical, and manufacturing) constraints with multidisciplinary solvers is diﬃcult. Each solver might make decisions with care for its target subset of constraints while potentially violating the rest of the constraints. (a) Arranging multiple solvers sequentially may work in some orders and fail in others, depending on what properties they preserve. Parallel composition requires combining solutions in ways that do not always preserve the properties either. (b) We propose a systematic two-phase approach (Section 3.4) to design space pruning (Section 4) and design space exploration (Section 5). The former invokes IP-solvers in an arbitrary order to cut out the infeasible design subspace without premature optimization. The latter navigates the pruned design space by ﬁxed-point iterations over FP-solvers. Section 3 addresses the question of how to divide a given collection of solvers into these two groups.

Lassifying Solvers And Constraints

In this paper we restrict our attention on solvers that re- duce material from a bounded domain in 2D and 3D space to generate designs. Our focus will be on the properties of constraints embedded within computational solvers to reason about their ordering in the workﬂows.

When all objectives and constraints cannot be handled by a single solver, current practice relies on a case-by-case domain-speciﬁc analysis to properly construct workﬂows.

Figure 7 (a) depicts examples of combining Unsweep and PareTO illustrated in Section 1.2. An arrangement that works in one case may not work in another.

In contrast, we present a systematic approach for or- ganizing solvers into workﬂows by ﬁrst classifying solvers

Into Two Fundamentally Diﬀerent Types:

1. Design space pruning solvers restrict the feasible de- sign space by pruning the subspaces that violate one or more design criteria. They are permutative, mean- ing that they can be called at the beginning of the design workﬂow in an arbitrary order. By directly op- erating on the design subspaces as ﬁrst-class entities, they postpone optimization to downstream solvers.

2. Design space exploration solvers simultaneously ex- plore the (pruned) design subspaces for optimized so- lutions. In this paper, we choose a well-studied Pareto

Front Tracing Approach (Pareto) To Eﬃciently

navigate the trade space of multiple objectives. Figure 7 (b) illustrates the framework. In this section we present some terminology and a classiﬁcation of design cri- teria based on which the solvers can be grouped into the above two classes. We formalize this two-phase approach in Section 3.4 and discuss how to implement each phase in depth with examples in Sections 4 and 5, respectively.

Esign & Performance Spaces

Throughout this paper, a ‘design’ Ωrefers to a com- putational model of a single designed artifact.

We Be-

gin the process by specifying a bounded design domain Ω0 ⊂Rd (d = 2 or 3) whose corresponding design space D is the collection of all ‘solids’, i.e., closed-regular semian- alytic pointsets in d−space,2 contained within the design domain. We use the notation D = P∗(Ω0) (read: “solid powerset” of Ω0). This deﬁnition is suﬃcient for geometric modeling of parts with homogeneous and isotropic mate- rial properties. We initially restrict our attention to this subclass of artifacts, as considered in classical solid model- ing . Our goal is to ﬁrst illustrate nontrivial challenges in designing with multiple solvers before introducing the additional complexity of multiple materials, heterogeneity, or anisotropy. However, the concepts introduced hereafter can (in principle) be generalized beyond solids.

The ‘performance’ of a given design Ω∈D is an n−tuple A(Ω) := (A1(Ω), . . , An(Ω)). Think of the performance space as a product space P := (F1 × . × Fn), where each Fi is a class of ﬁelds, i.e., each Ai ∈F1 is an integrable ﬁeld Ai(Ω) : Ω0 →Qi over the design domain Ω0 whose value at a given “query point” x ∈Ω0 is denoted by Ai(x; Ω) :=

(Ai(Ω))(X) ∈Qi. Examples Of Such Fields Are:

• binary-valued ﬁelds (Qi := {0, 1}), used to describe indicator functions of regions of interest within the

Design Domain Such As Non-Manufacturable Features

[7, 8] or regions requiring design correction . • integer-valued scalar ﬁelds (Qi := Z), used to charac- terize local topological properties of 3D printed parts [11, 12] or to classify atomic units of manufacturing in hybrid (combined AM and SM) manufacturing .

2A closed-regular set is deﬁned as a set that equals the closure of its interior (i.e., is homogeneously 3D) . Figure 8: (a) FP-solvers (e.g., FEA and manufacturability analysis) map an instance of the design space (i.e., a “design”) to an instance of the performance space (i.e., a “ﬁeld-tuple”). Predicates are deﬁned to decide whether the design is satisfactory with respect to constraints in terms of performance variables. An IP-solver modiﬁes the design until all constraints are satisﬁed.

• real-valued scalar ﬁelds (Qi := R), 3D vector ﬁelds

(Qi := R3), And Higher-Rank Tensor Fields Used To

represent distributed physical quantities such as dis- placement, velocity, stress, strain, and so on , or manufacturability measures .

Forward Problem In Physics & Kinematics

Forward problem solvers (FP-solvers) map a given de- sign instance to one or more performance ﬁelds, hence can be viewed as implementations of one or more maps Ai : D →Pi. The entire forward problem, solved by one or more FP-solvers, can be viewed as a single map from design space to performance space A : D →P, which has a unique outcome for a given design.

For example, consider a ﬁnite elements analysis (FEA) FP-solver that computes (discretized forms of) a displace-

Ment Field Uω:= A1(Ω) For Small Deformations Of A

given design Ω∈D due to boundary conditions such as restraints and external forces (Fig. 8).

T Is Also Im-

portant to compute the stress ﬁeld σΩ:= A2(Ω), which depends locally on displacement and material properties (e.g., the linear elasticity law). The vector/tensor values of solution ﬁelds probed at a query point x ∈Ω0 are de- noted by uΩ(x) = A1(x; Ω) and σΩ(x) = A2(x; Ω).

N

this case, both functions are zero outside the design, i.e., A1,2(x; Ω) = 0 if x ∈(Ω0 −Ω). FEA solves the weak form of the governing diﬀerential equation, discretized into a linear system (e.g., using hat functions) [KΩ][uΩ] = [f], where the stiﬀness matrix [KΩ] and external load vector [f] depend on the design shape and material properties, as well as boundary conditions. The equations are solved to obtain the discrete form of the displacement ﬁeld [uΩ] from which the discrete form of the stress ﬁeld [σΩ] is computed by linear operations.

Another example is accessibility analysis for machining (e.g., milling or turning) [7, 13]. For instance, consider an FP-solver for 3−axis milling simulation, which computes (discretized forms of) a volumetric measure of inaccessibil- ity as a ﬁeld µΩ(x) := A3(Ω) for a given design Ω∈D and machine tool parameters. This measure at a query point x ∈Ω0, denoted by µΩ(x) = A3(x; Ω), returns the pene- tration volume of the moving tool assembly T = (H ∪C), including the holder H and cutter C, into the stationary obstacles OΩ= (Ω∪F), including target form Ωand ﬁx- tures F.

The Solver Computes The Discrete Form Of This

ﬁeld (e.g., sampled at point clouds or voxels) [µΩ] as a convolution between the discrete forms of the indicator functions of stationary solids [1OΩ] and moving solids [1T ] using a fast Fourier transform (FFT) . The maximal set of accessible conﬁgurations – in this case, pure trans- lations in 3D for a ﬁxed orientation – is then obtained as

Ω(0), I.E., The Translations That Do Not

lead to undesirable collisions. The maximal removable vol- ume is obtained by sweeping the cutter with the maximal motion, i.e., RΩ:= sweep(MΩ, C). Its indicator function 1RΩ(x) = A4(x; Ω) can be viewed as a predicate for ac- cessibility, i.e., it returns 1 (resp. 0) if the query point x ∈Ω0 is (resp. is not) accessible. The discrete form of this binary ﬁeld [1RΩ] can also be obtained by thresholding FFT-based convolution of discrete forms [1MΩ] and [1C].

The Computations Performed By The Above Two Fp-

solvers (FEA and accessibility analysis) are abstracted by A := (A1, A2, A3, A4) that maps a given design to a “ﬁeld tuple” that represents analysis results. Figure 8 illustrates one instance of each such ﬁeld for a topologically optimized bracket—solution of a DARPA design challenge problem for TRAnsformative DESign (TRADES) program.

Feasible Design Subspaces & Predicates

Inverse problem solvers (IP-solvers), on the other hand, ﬁnd one or more designs that satisfy a given collection of functional requirements. Most IP-solvers employ an itera-

Tive Process To:

1. generate one or more valid candidate design(s); 2. perform analysis on the candidate design(s) to com- pute the performance(s) of interest (using one or more

Fp-Solver(S));

3. evaluate the performance(s) against given functional

Requirements; And,

4. if the requirements are not met, decide on the next generation of candidate design(s) based on the current evaluation and update rules.

The process is repeated until the requirements are met. The evaluation process (item 3) can be conceptualized as ﬁnite number of predicates deﬁned over the performance space as ci : D →{0, 1} for i = 1, 2, . . , n. Each predi- cate’s outcome ci(Ω) ∈{0, 1} is determined by means of a constraint imposed on the performance ﬁeld Ai(Ω) ∈Fi

(1)

These can be (in)equality constraints (Sections 3.2), which are common in physics-based design formulations as in TO, and set constraints (Section 3.3), which are ubiqui- tous in design under kinematics-based constraints such as packaging, assembly, and accessibility for manufacturing.

We can think of the performance criteria evaluation as a map c := (c1,c2, . . ,cn) : D →{0, 1}n i.e., c(Ω) is a binary string whose bits indicate whether a given design satisﬁes each of the criteria (Fig. 8). A design Ω∈D is called ‘feasible’ if it simultaneously satisﬁes all criteria, i.e., c(Ω) = (1, 1, . , 1). The feasible design subspace D∗⊆D

(2)

Here, (·)−1 denotes inversion of a mathematical relation.4 Unlike forward problems, inverse problems have non- unique, often inﬁnitely many, solutions (i.e., |D∗| > 1).

The feasible design space can also be deﬁned as the in- tersection of the feasibility halfspaces Hi := c−1

(1), Each

3Without loss of generality, we assume that every requirement depends on one performance ﬁeld only. If more than one ﬁeld is used in calculating a predicate, we tuple those ﬁelds into another ﬁeld. If more than one requirement is computed on one ﬁeld, we think of it as two copies of the same ﬁeld.

4Given f : X →Y , we deﬁne f−1(y) = {x ∈X | f(x) = y}. Note that in general, f−1(y) ⊆X is a set, i.e., f−1 : Y →P(X). It is a singleton set (i.e., has one element) if the function is bijective.

implicitly describing one of the design subspaces that sat-

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The idea of design space pruning (Section 4) is to pro- gressively cut out portions of the design space that vio- late any one of the criteria. Theoretically, pruning can be done in an arbitrary order—noting that intersections or unions of the halfspace in (3) are permutative. Compu- tationally, however, it is only possible if the design sub-

(1) Can Be Manipulated By Algorithms As

ﬁrst-class entities. Our goal is to understand under what conditions such manipulations are possible and how an entire design subspace can be represented. The existence (Proposition 1) and completeness (Proposition 2) of such representations are discussed in Section 4.

3.2. Global, Local, & Strictly Local Inequality Constraints The predicates introduced earlier are implemented in practice by testing whether a candidate design’s perfor-

We Classify

such constraints into three types; namely, global, local, and strictly local (in)equality constraints.

Global Inequality Constraints

It is common to have design criteria speciﬁed in terms of global constraints gi(Ω) ≤0, i.e., by deﬁning a predicate

(4)

in which gi : D →R is a function of the entire shape of the design Ω∈D, potentially in addition to ﬁxed external factors such as boundary conditions, manufacturing pro- cess parameters, packaging envelope, operating conditions (e.g., motion in assembly), etc.

The Constraint Is Often

evaluated in terms of a global property of an entire per- formance ﬁeld Ai(Ω) ∈Fi, e.g., as an upper/lower-bound on its maximum/minimum or its integral properties such ¯gi : D →R. For example, one can constrain the maximal displacement or maximal stress of a solid under external loads (Fig. 8) by using the following constraints in (4):

(6)

where δUB, σUB > 0 are constant upper-bounds on the magnitude of the displacement vector and stress tensor, captured by the constraints g1(Ω) ≤0 and g2(Ω) ≤0, 5Note that every equality constraint g(·) = 0 can be represented by two inequality constraints g(·) ≤0 and −g(·) ≤0.

respectively. One can in general use the p−norm of the ﬁelds for ﬁnite (but large) p ≥1, noting that maximum is a special case as p →∞. This is especially useful to smooth out the possible singularities (e.g., inﬁnite stress, due to stress concentrations).

Here is another example to constrain a design to be man- ufacturable via machining, using the accessibility analysis

(7)

where ¬1RΩ(x) = 1−1RΩ(x) is a negation. This constraint restricts the total volume of inaccessible regions RΩ⊆Ωc, obtained as the 1−norm of their indicator function, by an upper-bound VUB > 0.

Ocal Inequality Constraints

It is sometimes possible to deﬁne a predicate in terms of local constraints evaluated at a speciﬁc point in the design domain; for instance, using one of the following forms:

(9)

Note that the two alternative forms are diﬀerent by the “for all” and “there exists” quantiﬁers, which lead to dif- ferent global implications. Unlike the case with (4), here

Gi : (Ω0 × D) →R Is Field For A Fixed Design Ω∈D,

i.e., is also a function of the query point x ∈Ω0.

N

turn, the constraint is evaluated based on the probed value of the performance ﬁeld Ai(x; Ω) ∈Qi (generally, a ten- sor) at the query point. We denote this dependency by gi(x; Ω) = ¯gi(Ai(x; Ω)) where ¯gi : Qi →R. For exam- ple, the global displacement and stress bounds mentioned

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It is easy to verify that using (10) and (11) with (8) is equivalent to using (5) and (6) with (4) in this example. In general, local constraints gi(x; Ω) ≤0 used with “for all” or “there exists” quantiﬁers in (8) and (9) can be equivalently expressed as global constraints (stated inde-

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As another example, consider the accessibility analysis discussed earlier. Instead of constraining the total volume of inaccessible regions via the global constraint of (7), we can locally constrain the inaccessibility measure as:

Where Oω= (Ω∪F) And T = (H ∪C) Are The Stationary

and moving solids, respectively. µ0 > 0 is a small constant to provide allowance for numerical errors. The convolution

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where 1−T (x) = 1T (−x) is a reﬂection with respect to the origin, hence 1−T (x −x′) = 1T (x′ −x) is the indicator function of the moving object (i.e., tool assembly), trans- lated to the query point x ∈Ω0. The integral is nonzero at integration points x′ ∈Ω0 that belongs to the interference of the translated object with the stationary obstacles.

It is not always possible to convert global constraints to local constraints or vice versa, without deﬁning new per- formance variables, e.g., in terms of the norms of existing performance ﬁelds. We show in the next section that when Ai(x; Ω) is decidable independently of Ω∈D, the above two constraints lead to maximal/minimal feasible designs (in set-theoretic terms).

Strictly Local Inequality Constraints

An important special case of (8) occurs if the predicate is decidable without a priori knowledge of the design itself. In other words, the constraints can be evaluated purely from a knowledge of the query point’s position x ∈Ω and external factors, if any (e.g., a known rigid body mo- tion applied to the entire design). The predicate’s result can be obtained without knowing the overall shape of the design.

I (X), I.E., The

forward problem’s solution can be evaluated pointwise— emphasized by the overline notation. The corresponding

I (X)) ≤0 Is Hereafter Called A

strictly local (i.e., pointwise) constraint. The predicates in (8) or (9) in this case are dependent on Ωonly due to the

(17)

Hence, one can deﬁne pointwise predicates in this case:

(18)

We show in Section 4 that the pointwise predicate deﬁnes a point membership classiﬁcation (PMC) that implicitly determines the entire feasibility halfspace Hi = c−1

(1) Us-

ing its “representative” maximal/minimal feasible design

I

−1(1) using (16) or (17), respectively.

N This

paper, we restrict our attention to the former (“for all” quantiﬁer and maximal designs), since it lends itself to material reducing downstream solvers such as TO.

The physics-based constraints exempliﬁed earlier might reduce to pointwise constraints in rare examples—e.g., the stress tensor σ∗(x) for hydrostatic pressure in a liquid con- tainer at rest depends on the query point’s depth from the surface, but not on the container’s designed shape. Never- theless, pointwise constraints are ubiquitous in kinematics- based constraints that are central to applications ranging from assembly and packaging to manufacturing.

N The

next section, we present an important subclass of point- wise constraints that manifest as set constraints. 3.3. Converting Set Constraints to Inequality Constraints Many kinematics-based design criteria lead to con- straints expressed in the algebra of sets. A common form of set constraints is in terms of containment: Γ(Ω) ⊆E (for a ﬁxed envelope E ⊆Rd). The exact same constraint can be written in terms of non-interference: (Γ(Ω)∩O) = ∅ (for a ﬁxed obstacle O ⊆Rd), where Γ : D →P(Rd) is a set transformation and E := Oc (i.e., complement of O).

At a ﬁrst glance, these constraints appear to have a completely diﬀerent form than the inequality constraints described in Section 3.2 in the algebra of ﬁelds.

Here,

we show that set constraints can always be reformulated as (global or local) inequality constraints by virtue of de- scribing sets with their indicator functions. However, con- verting them to strictly local (i.e., pointwise) constraints is possible under certain conditions.

Global Or Local Set Constraints

For every solid Ω∈D, its indicator (i.e., characteristic)

(19)

Hence, every containment constraint is restated as an in-

(20)

i.e., using the standard form 1Γ(Ω)(x) −1E(x) ≤0. Recall from Section 3.2 that the above inequality con- straint can also be rewritten as a global constraint of the

(21)

Notice that we have not made any assumption on the properties of the pointsets Γ(Ω), E ⊆Rd. In most prac- tical scenarios, both are solids within a bounded domain, taken as the design domain Ω0 ∈D. In such cases, the properties of the mapping and envelope can be exploited to compute the maximum in (21) eﬃciently.

Moreover, if the set constraint (Γ(Ω) ∩Ec) = ∅is in terms of regularized intersection, it can be rewritten as a global (in)equality constraint in terms of the volume vol(Γ(Ω) ∩Ec] = 0 (or ≤0, for consistency), which, in

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Further, if Γ(Ω) is a rigid transformation of Ω, the inner product turns into a convolution of 1Ωand ¬1E over the conﬁguration space of motions .

Let us consider the inaccessibility analysis one more time. For manufacturing with precision requirements (e.g., for assembly/ﬁt), we can restrict the inaccessible regions Γ(Ω) := (Ω0 −Ω) −RΩto be completely contained within a tolerance zone E ⊆Rd, hence formulating the problem

As A Set Constraint Γ(Ω) ⊆E, Where:

Γ(Ω) = (Ω0 −Ω) −sweep(MΩ, C).

(24)

For translational motions MΩ⊆Rd, the sweep in (24) can

(25)

The maximal collision-free motion is obtained as the complement of the C−space obstacle, which is in turn computed by Minkowski operations (for translational full-

(27)

where T = (H ∪C) represents the tool assembly, includ- ing the holder H and the cutter C.

T Is Understood

that Minkowski sums in (26) can be obtained from the 0-superlevel set of the convolution ﬁelds of the participat-

(28)

where ∗is the convolution operator in Rd and sign(x) = 1 (resp. 0) if x > 0 (resp. x ≤0) is the sign function. The latter is needed to convert the real-valued convolutions to binary-valued indicator functions. See for a proof.

In summary, set constraints, as deﬁned in this section in terms of containment or non-interference, can always be restated as global or local inequality constraints deﬁned in Section 3.2. In the next section, we show the conditions under which set constraints can be converted to strictly local (i.e., pointwise) constraints, to enable design space pruning in Section 4.

6The operators (·)c, ∪, ∩, −, ⊕, ⊖are all regularized to ensure their algebraic closure within the design space D . The caveat is that collision-free lower-dimensional motions (e.g., of a peg inside a hole with no clearance) would be pruned from collision-free motions.

7In general, Y = (X1 ⊕X2) ⇌1Y = sign(1X1 ∗1X2) .

Set Constraints As Pointwise Constraints

Depending on the properties of Γ : D →P(Rd), the in- equality constraint 1Γ(Ω)(x)−1E(x) ≤0, used in global or local forms of (20) and (21), respectively, may or may not be restated as a strictly local (i.e., pointwise) constraint.

Our goal in this section is to articulate the conditions un- der which this is possible. To enable pointwise formulation, we need to eliminate the dependency of the PMC for Γ(Ω) on Ωso that 1Γ(Ω)(x) on the left-hand side of the inequality constraint in (20) depends only on the query point x ∈Ω0 and the ﬁxed envelope E ⊆Rd.

This Can Be Done If The Set Trans-

formation Γ is itself a pointwise transformation, meaning that it can be deﬁned by extending a transformation of 3D points γ : (Rd or Ω0) →P(Rd) to a transformation of 3D pointsets Γ : D →P(Rd) by simply applying the former to every point of the pointset and unifying the results:

(29)

Note that γ(x) is itself a pointset, not a point, to capture the most general case. For example, it can be a curve seg- ment or surface patch representing the 1D or 2D trajectory of a point under a given one- or two-parametric motion, respectively.

The above refactoring is possible for many important applications. For example, if the design has to move (when deployed in assembly) according to a known motion set M ⊆SE(3) without exiting a safe region of space E ⊆Rd that contains no obstacles , the above constraint can

(30)

is the sweep of the designed part as it travels by the given motion (known a priori). In this case the sweep indicator function 1Γ(Ω) can be directly obtained as follows:

(31)

In other words, a PMC test for Γ(Ω) can be obtained by applying the inverse motion M −1 = {τ −1 | τ ∈M} to the query point and checking if it intersects the design. The inequality constraint in (21) can thus be computed rapidly by sampling query points in the design domain and testing intersections of their inverse trajectories with the design . Importantly, the computation for one point does not need one to have explicit knowledge of the results for other points. Other than perfect parallelization (e.g., on GPU), this property enables pruning the design space – leading to development of the Unsweep solver discussed earlier – before optimizing the design for other criteria.

In general, for pointwise transformations as in (29), the

(32)

i.e., Ωremains within E after a Γ−transform iﬀall points inside it remain within E after a γ−transform. Note also that inequality constraint in (20) can now be rewritten in

(33)

As with other inequality constraints, not every global or local set constraint can be converted to a pointwise set constraint. For example, the toleranced accessibility con- straints Γ(Ω) ⊆E for Γ(Ω) := (Ω0 −Ω)−sweep(MΩ, C) in (24) cannot be evaluated pointwise, because the maximal collision-free motion MΩ= (Ω⊕(−T))c depends on the global shape of Ω, unlike the case with the ﬁxed motion in the earlier example with Unsweep.

Esign For Heterogeneous Constraints

We conclude this section by a fairly general formulation of a design problem subject to n ≥1 heterogeneous (e.g., kinematics- and physics-based) constraints.

Without loss of generality, let nC = nG +nL +nP where 0 ≤nG, nL, nP ≤nC are the number of global, local, and strictly local (i.e., pointwise) constraints, respectively. All constraints, including set constraints, are expressed as in- equality constraints for uniformity. The design problem is to identify the feasible design space D∗= c−1(1), stated

(34)

We assume that none of the gi(x; Ω) ≤0 can be simpliﬁed

Into One Of G∗

the P-, G-, and L-subscripts for various notions related to pointwise, global, and local constraints, respectively; for

P (1) Is The

design subspace that is feasible with respect to pointwise constraint alone, and so on. We solve the design problem

(35)

In Section 4, we solve the above problem by computing

P := Max D∗In The Partial Ordering Of

designs via set containment.

N

most cases, one can at best generate a ﬁnite sample of feasible designs that are superior in some way.

P And Ω∈N(Ω) In The Pruned Design

space,8 within which no other design is superior to Ωwith respect to all objective functions f1, f2, . . , fnO : D →R.

Minimize Fj(Ω), For 0 < J ≤No,

subject to constraints in (36).

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The objective functions deﬁne another partial ordering over the pruned design space, whose maximal elements

We Seek, I.E., D∗

dom := max D. In Section 5, we solve the above problem by iterative op- timization guided by TSF. The TSF is deﬁned with respect to global objective functions fj(Ω) and global constraints gi(Ω) ≤0 and is penalized/ﬁltered using local constraints gi(x; Ω) ≤0. Since global optimization (N(Ω) := D∗

P) Is

NP-hard , we settle for local optimality.

Esign Space Pruning

In this section, we present a methodology to solve the phase 1 problem formulated in (35) of Section 3.4. Us- ing the terminology and results of Section 3, we show how the design space can be pruned with respect to pointwise constraints (including set constraints) without premature optimization.

We Illustrate The Process Using Examples

from kinematics-based constraints that are common in as- sembly, packaging, and manufacturing.

Existence & Completeness Of Maximal Designs

The following results on the existence and uniqueness of maximal pointsets and their informational completeness as a representation for entire feasible design spaces are central to design space pruning.

Proposition 1. (Existence and Uniqueness) For every

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The maximality is in terms of set containment, i.e., every satisfactory design is contained in the maximal element: Ω∈Hi ⇒Ω⊆max Hi.

8Technically, N(Ω) is an open set in the induced Hausdorﬀtopol-

Ogy Of D∗

P = P∗(Ω∗) (all solid subsets of the maximal element). Proposition 2. (Completeness) For every strictly local

I = Max Hi, I.E.,

every solid subset of the maximal element is also feasible: Ω⊆max Hi ⇒Ω∈Hi. In terms of predicates, the design subspace Hi = c−1

(1)

(which satisﬁes (16)) can now be represented by a single

(1) (Whose All Points Satisfy (18)). The

maximal solid is thus a complete representation of the fea- sibility halfspace as the collection of all of its solid subsets

I Deﬁned By (38) Or (39) Contains All Points

that satisfy the constraints.

I Of The Maximal Set Satisﬁes

the constraint, because all of its points satisfy the constraint independently of the global shape. 3. Conversely, every feasible solid Ω∈Hi is the subset

Of Ω∗

i , because it only includes points that satisfy this constraint independently of the global shape. Note that the constraint’s independence of the shape of Ωis crucial for this to hold.

For Global Or Local Con-

straints with dependency on Ωitself, attempting to write a PMC similar to (38) leads to a circular deﬁnition where the right-hand side depends on the set itself. For example, the (global or local) constraints given in Sections 3.2.1 and 3.2.2 on FEA and printability analyses do not lead to max- imal elements because their constraints gi(Ai(x; Ω)) ≤0 depend on particular design instances. It does make sense to deﬁne the maximal set of a feasible space (e.g., using the set-builder deﬁnition in (39)) in a way that it depends on a particular instance of that space. On the other hand, the set constraints of Section 3.3.2, such as containment under a prescribed motion, do give rise to maximal elements, as we elaborate with examples in the following sections.

The above reasoning does not take topological regular- ization into account—there is no reason for a maximal set obtained via (38) or (39) to be a solid, hence it may not it- self be a valid design. However, we show that there always exists a valid maximal element obtained by regularizing

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where k and i are the topological closure and interior op- erators, respectively. Proposition 3. (Design Space Pruning) Given a number

I (X) ≤0 For I = 1, 2, . . . , Np, The

feasible design space D∗, deﬁned by intersecting all feasi-

P = Max D∗

that satisﬁes the uniqueness and completeness properties,

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in which the intersection/conjunction operators need to be regularized. To see why this is true, note that any query point’s membership in an (unknown) feasible design can be tested against all m constraints independently of other points’

I (X) ≤0 For I = 1, 2, . . . , Np, The Point

can (but does not have to) be included in the design, for it to be feasible. The feasible design is hence a subset of all points that satisfy all pointwise constraints.

The above result enables computing on design subspaces as ﬁrst-class objects. Computationally, the feasibility half- spaces are represented uniquely by their maximal elements.

The pruning of halfspaces (abstract operation) is imple- mented by intersecting maximal elements, i.e., conjunct- ing point membership tests deﬁned by pointwise constraints (concrete algorithm) in an arbitrary order.

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Figure 9 illustrates design space pruning via (3). Figure 9: Design space pruning can be abstracted by intersecting the design space with feasibility halfspaces. This is not a computable operation in general. However, for pointwise constraints, it can be computed by intersecting maximal elements in the physical space.

Next, we consider maximal elements for set constraints in Section 4.2, and provide examples from real-world en- gineering problems in Section 4.3, 4.4, and 4.5.

Aximal Pointsets For Set Constraints

Many design requirements, especially those that relate the interaction of moving shapes, may be expressed as set constraints of the form Γ(Ω) ⊆E, where Γ : D →P(R3)

Is A Set Transformation. For Example,

• In packaging and assembly problems, the part’s shape is often designed so that it is restricted to remain

Within A Speciﬁed Envelope While Moving According

to a prescribed motion. • When designing part to be machined using a given tool that moves in the presence of obstacles (e.g., the part itself and ﬁxtures), the surface of the part has to be accessible without collisions with the obstacles.

We will address examples of these problems in the fol- lowing sections.

The Common Theme To Many Motion-

based set transformation Γ : D →P(R3) is that they can re-factored via (29) via a pointwise transformation γ : Ω0 →P(R3) that depends on the motion. The maxi- mal pointset that satisﬁes the containment Γ(Ω) ⊆E or non-interference test (Γ(Ω) ∩Ec) = ∅is deﬁned implicitly

(45)

where the set operators are regularized, as before. Note that γ−1 may not be a function, because γ is not neces- sarily invertible (see footnote 4).

Here are a few classical examples from solid modeling: • For one-parametric sweep Γ(Ω) := sweep(M, Ω), one

Has Γ(X) = Mx Where M = M(T) ∈Se(3) Is A Contin-

uous one-parametric set of motions for t ∈[tmin, tmax].

The Maximal Shape That Satisﬁes Γ(X) ⊆E Is Given

by an unsweep unsweep(M, E) .

• For Minkowski Sum Γ(Ω) := (Ω⊕B), One Has Γ(X) =

(x + B) where B ⊂R3 is typically a solid.

Maximal Shape That Satisﬁes Γ(X) ⊆E Is Given By

a Minkowski diﬀerence with (E ⊖(−B)) . • For general dilation (which subsumes the above two) with general rigid motions, the maximal shape is given by general erosion .

• For non-rigid (but pointwise pre-determined) defor-

Mations, The Maximal Shape Is Obtained By Its Pmc

in terms of the pointwise displacement function. Procedure. Propositions 1 through 3 suggest a system- atic procedure to prune the design space, i.e., reduce the

P = D ∩(H1 ∩H2 ∩· · · ∩Hnp) For

kinematic criteria expressed in terms of pointwise set con-

Straints:

• Step 0. Initialize the feasible design space with the

P ←Ω0.

• Step 1. Express the set constraint that outlines one of the conditions for a given design Ω∈D to be feasi- ble in the form Γ(Ω) ⊆E. Check if it can be restated as a pointwise constraint γ(x) ⊆E as in (32).

• Step 2. Formalize the forward problem in terms of the PMC test for the maximal element obtained from the pointwise constraint as prescribed by (43).

• Step 3. Invoke an IP-solver for the inverse problem,

Tation Of) The Maximal Element Ω∗

i in (44). • Step 4. Prune the design space by intersecting the

I ) Is A Smaller Maximal Element Repre-

senting a pruned feasible subspace. • Repeat steps 1–4 for all pointwise set constraints. Notice that the above procedure can be applied to diﬀerent constraints via independent invocation of IP-solvers in an arbitrary order.

Importantly, the IP-solver in step 3 can be implemented in two fundamentally diﬀerent ways to obtain either an implicit representation (i.e., using PMC test in (43)) or an explicit representation (i.e., using inversion in (45)) of the

Maximal Element:

• If we have access to an IP-solver that computes an ex- plicit representation (e.g., B-rep) of the inverse trans- formation γ−1(E), we can directly intersect it (using

Any Cad Kernel) With The Design Domain To Obtain

the maximal element as prescribed by (45). • If we have access to a FP-solver that computes (ex- plicitly or implicitly) the forward transformation γ(x)

For A Given Query Point X ∈Ω0, We Can Compute An

approximate representation (e.g., point cloud or vox-

Invoking The Fp-Solver To Pmc-Test Them Using

(45); keep the ones that pass the test and discard

The Ones That Do Not; And

3. (optional) use adaptive local re-sampling around the points that passed the test to obtain a better approximation.

Because of the independence of pointwise test, invo- cation of the FP-solver for diﬀerent query points can be done with perfect parallelization.

Let us apply this procedure to a few examples.

Pruning For Containment Of Moving Parts

Consider the latch design problem introduced earlier in Section 1.2, where the goal is to design a car hood latch that remains within an envelope E ⊆Ω0 while moving according to a motion M ⊆SE(3).

Step 1. Every feasible latch design Ω∈D must satisfy the set constraint Γ(Ω) ⊆E where Γ(Ω) := sweep(M, Ω); i.e., the swept volume by the latch after being transformed by all conﬁgurations τ ∈M (including any combination of translations and rotations, parameterized or otherwise) remains within the envelope.

The Sweep Is A Pointwise

transformation, i.e., can be computed as the union of all

Τ∈M Τx, Which Represents The Trajectory

traced by the query point x ∈Ω0 along the prescribed motion. Hence, the containment constraint be tested in a pointwise fashion by γ(x) ⊆E.

Step 2. Using the deﬁnitions in (43), we construct a PMC test for the maximal shape in the design space that satisﬁes

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The forward problem involves following the trajectory, ei- ther exactly or approximately (e.g., by sampling), and testing whether it remains entirely within the envelope.

Step 3. The dual properties of the FP- and IP-solvers (i.e., Sweep and Unsweep) can be leveraged to con- struct an exact or approximate representation of Ω∗

, As

illustrated in Fig. 10. If we have access to an Unsweep

= Unsweep(M −1, E) ∩

Ω0. However, if we only have access to an eﬃcient Sweep solver, we can still compute an approximate representation

Of Ω∗

1 using the PMC test in (46) for a suﬃciently dense sample of query points and retaining the points whose for-

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The invertibility of rigid transformations M 7→M −1 is key to eﬃcient direct implementation of Unsweep. Figure 10: unsweep(R, E) is the largest set that remains within the square envelope E shown in (a) while moving by a 21◦clockwise rotation R around a ﬁxed pivot , i.e., without violating the con- tainment constraint as depicted in red in (b). The unsweep can be computed in general by intersecting all moved instances of the enve- lope with an inverse motion .

The remaining steps 4 and 5 are straightforward.

Ω∗

1 can be sent as the initial design to design space ex- ploration (Section 5) via TO or any other downstream ma- terial reducing IP-solver. It is guaranteed that every valid

That Is The Output Of The Downstream Ip-

solver, no matter how complicated, will continue to satisfy the set constraint sweep(M, Ω) ⊆E.

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.