Optimizing micro-tiles in micro-structures as a design paradigm
Pablo Antolin, Annalisa Buffa, Elaine Cohen, John F. Dannenhoffer, Gershon Elber, Stefanie Elgeti, Robert Haimes, Richard Riesenfeld
OOptimizing micro-tiles in micro-structures as a design paradigm
Pablo Antolin ´Ecole Polytechnique F´ed´erale de Lausanne, Institute of Mathematics, Lausanne, Switzerland
Annalisa Buffa ´Ecole Polytechnique F´ed´erale de Lausanne, Institute of Mathematics, Lausanne, Switzerland and Istituto di MatematicaApplicata e Tecnologie Informatiche ’E. Magenes’ (CNR), Pavia, Italy
Elaine Cohen
Department of Computer Science, University of Utah, SLC, Utah, USA
John F. Dannenhoffer
Mechanical & Aerospace Engineering, Syracuse University, Syracuse, NY, USA
Gershon Elber
Department of Computer Science Technion, Israel Institute of Technology, Haifa 32000, Israel
Stefanie Elgeti
Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Aachen, Germany
Robert Haimes
Department of Aeronautics and Astronautics, MIT, Cambridge, Massachusetts, USA
Richard Riesenfeld
Department of Computer Science, University of Utah, SLC, Utah, USA
Abstract
In recent years, new methods have been developed to synthesize complex porous and micro-structured geometry ina variety of ways. In this work, we take these approaches one step further and present these methods as an efficaciousdesign paradigm. Specifically, complex micro-structure geometry can be synthesized while optimizing certain propertiessuch as maximal heat exchange in heat exchangers, or minimal weight under stress specifications.By being able to adjust the geometry, the topology and/or the material properties of individual tiles in the micro-structure, possibly in a gradual way, a porous object can be synthesized that is optimal with respect to the designspecifications. As part of this work, we exemplify this paradigm on a variety of diverse applications.
Keywords:
Analysis, heterogeneous materials, topological optimization, porous geometry.
1. Introduction
Recently, in [11, 17], methods and algorithms for theprecise construction of micro-structures using functionalcomposition [9, 10] were proposed. In that approach, thedesign of the macro-shape and the micro-structures of aporous geometry are decoupled. A parametric form of a(typically periodic) micro-tile M is specified as some com-bination of curves, surfaces, and/or trivariates while themacro-shape T is also specified as a parametric trivariatefunction T : D ∈ IR → IR . See Figure 1. D , the domainof T , is populated with tiles M i , only to compute the final result as the function composition T ( M i ) , ∀ i .The approach of [11, 17] has a very simple set of inputs,namely: • A micro-tile M , as some combination of para-metric curves C ( t ) = ( c x ( t ) , c y ( t ) , c z ( t )), surfaces S ( u, v ) = ( s x ( u, v ) , s y ( u, v ) , s z ( u, v )) and trivari-ates T ( u, v, w ) = ( t x ( u, v, w ) , t y ( u, v, w ) , t z ( u, v, w )).Without loss of generality, we assume that M is con-fined to a designated volume, specifically a unit cube. M is typically periodic in the sense that the d min facesare C -continuous with respect to d max , d = x, y, z , Preprint submitted to SPM 2019 July 5, 2019 a r X i v : . [ c s . C G ] J u l a) M (b) D (c) T (d) T ( M ) Figure 1: The micro-tile M in (a) populates the domain D , in (b),(2 × ×
2) times. D is the domain of trivariate T shown in (c). Alsopresented in (c) are the (2 × ×
2) composed tiles T ( M ). Finally,(d) shows only the composed tiles T ( M ) and may even be C k -continuous, k > • A trivariate parametric deformation macro-function T ( x, y, z ) : D ∈ IR → IR . • ( n x , n y , n z ): the dimensions of enumerations in T , in( x, y, z ) of the micro-tile M .Algorithm 1 summarizes this entire process. Algorithm 1 Micro-structures synthesis using func-tional compositionInput : M : a micro-tile consisting of curves and/or surfacesand/or trivariates; T : a trivariate parametric deformation macro-function T ( x, y, z ) : D ∈ IR → IR ;( n x , n y , n z ): the dimensions of the grid enumeration oftiles M in the domain of T ; Output : MM : A micro-structure of ( n x , n y , n z ) tiles in a 3D griddeformed to following the macro-shape of T , via func-tional compositions, as T ( M ); Algorithm : MM := ∅ ; for k = 1 , n z do for j = 1 , n y do for i = 1 , n x do M ijk := M positioned at ( i, j, k ), in D ; M ijk := T ( M ijk ); // via func. composition MM := MM ∪ {M ijk } ; end for end for end for return MM ;The result is a precise layout of individual tiles that(continuously) follows the macro shape. However, in D ,the parametric domain of T , all the placed tiles M ijk areof the same geometry (up to rigid motion). Clearly, inas-much as T is rarely isometric, T ( M ijk ) can be arbitrarilydeformed, which is where this effort begins.One application that demonstrates these compositionbased parametric micro-structures is the design of a heat sink [3]. Following the goal of using maximimal surfacearea to optimize the dissipation of heat while minimizingthe volume of material, the synthesis of the geometry canbe achieved with relative ease through exploiting hierar-chical parametric micro-structures. See Figure 2.Striving to control the shapes of the tiles in Euclideanspace serves as a first motivation for this work. Some gen-eral design specifications can be imposed on the object ofdesign in the form of strength, weight, heat or electricalconductivity, etc. One can then consider the geometryand the topology of individual tiles, as well as some inte-rior (materials) properties, as degrees of freedom in someanalysis or optimization process, a typical part of a designcycle.The rest of this work is organized as follows. In Sec-tion 2, previous efforts in synthesizing micro-structuresare surveyed. Section 3 presents the necessary buildingblock to enable this proposed design paradigm of micro-structures. In Section 4 some examples are considered ina variety of engineering applications. Finally, we concludein Section 5.
2. Previous work
In recent years, the interest in porous geometry andmicro-structures, on one hand, and in heterogeneous ma-terials on the other, has been on the rise. One major moti-vation for 3D representations stems from the new abilitiesintroduced by additive manufacturing to fabricate suchstructures [14]. Porous geometry, micro-structures, andheterogeneous materials, all enabled by modern additivemanufacturing technologies, are finding applications in avariety of fields from medicine and biology [2], throughmechanical and aero-space engineering to electric engineer-ing [25] and material science [1].The authors in [5] perform optimization of micro-structures towards multi-scale modeling. A set of parame-ters is identified at the micro-scale level that govern prop-erties of the model like the shape, stress, strain, etc. Thefinite element method is used in the analysis step. Evo-lutionary schemes, which are based on genetic algorithms,are used to search for optimal parameter values. Althoughthe NURBs representation is used for the shape of thestructures, such an approach lacks precision.In [7] the authors present a framework for modeling het-erogeneous objects using trivariate B´ezier patches. TheB´ezier patches have two sets of coordinates. The first setof three coordinates ( x, y, z ) prescribe the shape of theobject, while the rest of the coordinates specify the mate-rial composition of the object. While this allows for con-struction of a wide range of structures with functionallygraded materials (FGM), it only admits a single level ofdetails, and is limited only to B´ezier trivariates. Modelingof micro-structures, for instance, is beyond the scope ofthis work.The authors in [26] adopt principles from stochas-tic geometry [15] for designing porous artifacts. While2a)(b) (c) (d)
Figure 2: A heat sink design using a hierarchical micro-tile (a) and a macro-shaped twisted ring (b), yielding the heat sink in (c), with largetotal surface area. (d) shows a zoomed-in view on the top right portion of (c). Following the introduction of hierarchical micro-structures in[17]. such an approach is suitable for designing random micro-structures, for example toward bone tissues, it is less fa-vorable for applications requiring a high level of precisionlike, for instance, a wing of an aircraft. Also, the approachdoes not address the question of ensured connectivity ofthe porous structures. The authors in [30] use Voronoitessellations to also generate random porous structures ofthree kinds: porous geometries with intersecting fractures,interconnected tubes and fibers. A set of points is ran-domly sampled in the space which gives rise to Voronoitessellations. Offsets are then computed for the edgesof the tessellations to generate the three kinds of micro-structures. The authors’ aim at fluid flow analysis whilethe presented approach is limited to constructing geometrywhich is piecewise linear or cylindrical.In [20] the authors use implicit surfaces for modelingmicro-structures. Their approach allows for design of reg-ular as well as irregular structures that are amenable togeometric operations such as blending and deformation.Such an approach does not readily support the creationof FGM objects. Further, this implicit approach does notguarantee connectivity in the case of irregular (random)structures.In [2] the authors propose a method for creating scaf-folds as support structures for tissue engineering. The scaf-folds, once fabricated out of some biodegradable or biore-sorbable material, are seeded with (biological) cells andprovide support and shape to tissue during its growth. Inthis approach, the scaffolds are modeled as porous micro-structures using a polygonal representation. The methoddoes not generalize to freeform spline geometry so FGMobjects are not supported.In [29] the authors propose a method for designing meso-scopic structures using trusses. The output of the system isin the form of triangles in STL format. Since this methodis tailored to trusses as the basic building blocks of thestructure, the scope of application is limited. The authorsmention filling the volume of trivariates with tiles, thoughno details are provided. The authors in [6] also propose aframework for designing additive manufactured mesostruc-tures. Their method is based upon the process-structure-property-behavior model. The basic building element is an octet truss which is represented parametrically. Ex-tensions for support of other types of elements are notaddressed.In [18] the authors propose a method to design cellularstructures geared towards additive manufacturing. Thecellular structure is achieved through an adaptive trian-gulation of the interior of the solid, with finer tetrahe-dra along the boundary of the solid. Their method alsosupports a dual construction obtained from the Voronoidiagram of the triangulation. The approach does not gen-eralize to freeform geometry.In [16] the authors propose a modeling primitive basedon a generalized cuboid shape, which is referred to as ablock. The design of a complex object proceeds by layingout blocks, which are then connected to form the basicshape of the object to be modeled. A control mesh isthen extracted from the faces of the blocks, which allowsparameterization of the surface. While this provides for asimple and elegant modeling approach, it does not allowthe two stage methodology for the design of micro andmacro structures.Some of the above work either ignores or is not capableof supporting analysis or optimization applications overthe synthesized geometry. In others, analysis is feasiblewhile it is unclear how the results of the analysis or opti-mization stages can be fed back into the geometry in orderto enhance the designed shape to complete the design andanalysis cycle. In this work, we propose a new paradigmfor the design of precise FGM micro-structures and porousgeometry using functional composition. By parametricallycontrolling the geometry, topology and materials of indi-vidual micro-tiles, one is provided with a tight link be-tween the design and analysis or optimization stages, andhence establish a design framework of such structures. Thenext section, Section 3, we present the main concept be-hind the proposed paradigm.
3. Micro-structures Synthesis as a DesignParadigm
Recalling Algorithm 1 and following [11, 17], the con-struction of micro-structures involves three main steps:3 igure 3: Parametrically varying wall thicknesses/tubes diametersin a solid tile M with a single boundary wall on its right side (top).Parametrically varying diameters of hollowed tubes in a tile M witha pair of fixed thickness boundary walls (bottom).
1. Specifying the macro shape parametrically.2. Paving a basic micro-tile M in the domain of themacro-shape or the deformation map T as { M ijk } .3. Functionally composing the tiling created in step 1, { M ijk } , into the deformation map T , to obtain therequired micro-structure T ( M ijk ).Assuming the tile M is a (trimmed) trivariate or a setof such (trimmed) trivariates, and following this pavementprocess, isogeometric analysis can be immediately appliedto the structure, as is done in [17], for example. Then,the results of the analysis and/or optimizations can be feddirectly to the next geometry synthesis cycle in which theparameters of the geometry/material properties, etc., aremodified to follow the provided constraints and analysisor optimizations. In order to specify material propertiesand other fields, the control points for the trivariates areextended to IR k , k ≥
3. The first three dimensions are usedfor the geometry, whereas higher dimensions are optionallyused for holding scalar, vector and tensor properties, suchas materials.Let P = { p , p , ..., p n } be a set of n parameters for afamily of geometric shapes, topologies and FGM distribu-tions, etc., of a tile M . Typically, these parameters will beconstrained so critical properties such as the sign of theJacobian, will not be affected. Figure 3 shows differenttiles with parametric control over the wall thicknesses ofboth solid and hollow M , and illustrates parametric con-trol over the diameters of the tubes in a tile.Algorithm 1 can now be modified to accommodate black box optimizations or analysis tools to manipulate themicro-structure. See Algorithm 2.In Algorithm 2, the geometry is synthesized using amulti-parameters family of micro-tiles. The optimizer O sets the parameters (i.e. wall thickness or pipe diameter)at iteration i , to affect only the synthesized geometry atiteration i + 1. While Algorithm 2 hints at parametersthat control the geometry, the parameters can also controlmaterial properties like heat conductivity.For the first pass of the optimization, Algorithm 2 needsto synthesize an initial micro-structure so the optimizer O has something on which to operate. This initial micro-structure is application dependent and in Section 4 wepresent several such applications. Algorithm 2 Micro-structures synthesis using func-tional composition and optimizationInput : M ( P ): a parametric micro-tile consisting of curves and/orsurfaces and/or trivariates; T : a trivariate parametric deformation macro-function T ( x, y, z ) : D ∈ IR → IR ;( n x , n y , n z ): the initial dimensions of the grid enumeratingtiles M in the domain of T ; O : an optimizer - a black box that optimizes the synthe-sized micro-structure; Output : MM : A micro-structure of ( n x , n y , n z ) tiles deformed tofollowing the macro-shape of T via functional composition T ( M ( P )) as influenced by optimizer O ; Algorithm : P := init parameters for MM ; while Not Optimized do MM := ∅ ; for i = 1 , n x do for j = 1 , n y do for k = 1 , n z do P ijk := geometry/topology/material param-eters of tile ijk , as prescribed by P ; M ijk := M ( P ijk ) at position ( i, j, k ), in D ; M ijk := T ( M ijk ); // via func. composition MM := MM ∪ {M ijk } ; end for end for end for P := solve O ( MM ); end while return MM ;The presented scheme provides for significant freedom tomodify individual micro-tiles while preserving the continu-ity of the the entire arrangement. If a tile is modified inthe domain of the deformation macro-function, and its ad-jacent micro-tiles are updated accordingly to preserve thecontinuity, the outcome will preserve continuity as well(assuming the deformation macro-function is sufficientlycontinuous). Further, the continuity of the geometry mustbe preserved as well as the continuity of properties such asmaterial. Finally, the modified topology should not affectthe continuity. For example, it is infeasible to modify abranching tile from 2 branches to 4 branches without alsomodifying adjacent tiles to retain continuity properties.
4. Applications and Examples
In this section we present four different design opti-mization application of micro-structures. In Section 4.1micro-structures are optimized toward heat exchangers.4 igure 4: This contrasts a traditionally used element on the left witha simpler tile element on the right that can be easily generated viaadditive manufacturing. Note that the red and blue colored surfacesrepresent those whetted to the individual fluid channels, where thetan color refers to metal.
Section 4.2 considers the question of solid heterogeneousrocket fuel design. In Section 4.3 a wing design based onmicro-structures is considered, and finally, in Section 4.4,local thermal control, is optimized in plastic extruders.
Although often not visible from the exterior of a me-chanical system and not the focus of widespread atten-tion, heat exchangers are used universally for heating andcooling functions. Common examples of heat exchangersinclude radiators, oil coolers, refrigeration/air condition-ers/heat pumps, and etc. They occur in a variety of typeswhose forms are typically driven by applications and tra-ditional manufacturing techniques.Typically, heat exchangers are selected from a cata-log of rectilinear or cylindrical prismatic shapes, thusone is forced to design around the existing standard pre-manufactured shapes. This can lead to awkward, ineffi-cient applications because the design must be adapted toconform to and accommodate the catalog available heatexchanger shapes. The designer is required to adjust theflow so that the conditions at the inlet of the heat ex-changer can result in good heat transfer. Additive man-ufacturing opens the opportunity to design custom heatexchangers specific to the cavity or envelope that natu-rally occurs in an emerging design.The simplest, inexpensive and most common heat ex-changers are manufactured by taking a sandwich of corru-gated metal, a metal sheet and then corrugated metal (at90 degrees to the first corrugated layer), another sheet andthen repeating. This suite of plies is then welded togetherand headers added to create a closed loop for one of thefluids. If one were to generate a single tile of this device itwould look like the left-hand image seen in Figure 4. Forthe rest of the discussion we will be using a simpler repeatpattern that is easily constructed by additive manufactur-ing and can be seen on the right side of Figure 4.Figure 5 depicts the composition of the tile seen in theright side of Figure 4 into a duct with changing cross-sections. In this case the spacings in the duct are changing,which is driven by the the knot sequence of the geometrythat defines the duct ( T orig ). Figure 5: This sequence of images illustrates the flexibility affordedby a parametric heat exchanger design. The entire available cavityor duct can be used for the device and the number of passages ofeither fluid can be adjusted (as well as the spacing). The upper leftimage is the baseline where the upper right has increased the numberof blue passages. The lower left image show the same duct with alarger number of red passages. The lower right image is simply morepassages for both fluids.
An effective heat exchanger maximizes the heat transferover the entire device. This is not an isolated geomet-ric problem but is a complex heat transfer problem relat-ing the geometry, metal properties, thermal gradients andflow characteristics of the device. A low fidelity, lumped-parameter heat exchanger model has been constructed forthis problem that can output trivariate metal thicknesses,hot-to-cold area ratios, and relative tile sizes throughout T orig . The model takes into consideration the physicallaws in each hot and cold channel (including temperatureand pressure of the appropriate fluid), deals with mass andenergy conservation while using a convective heat transfermodel. This is accomplished before the geometry of theheat exchanger is composed.The first step in building the device is to reconstruct theduct so that its knot sequence reflects the relative spac-ings of the tiles ( MM ) in physical space ( T orig ⇒ T ). Thisis done so that each tile is generated with the appropri-ate size. Algorithm 3 is then followed, where it shouldbe noted that each tile is individually constructed duringthe composition. The tile’s parameters are sampled fromthe scalar trivariate fields, so that after the constructionis complete the geometry matches up at the individual tileinterfaces. This allows for the building of a coherent struc-ture (by sewing) that can be represented as a geometric solid .Figure 6 shows the results of Algorithm 3 on a simplerectilinear outer shape with a small number of tiles. Theleft side of the figure show all of the individual tiles usedin the composition ( M ijk ), where the right-hand side ofFigure 6 shows the composed result driven by the spac-ings and thicknesses output from the low fidelity, lumped-parameter heat exchanger model. This application concerns the use of accelerants and re-tardants mixed with the propellant in a solid rocket engine.Mixing the fuel is considered in order to control and planthe burn rate and thereby the expected thrust. The prob-lem given is to match a prescribed thrust profile and the5 igure 6: The left side depicts the individual parametric tiles in their pre-deformed unit representation. The right side image shows thecomposed resulting heat exchanger in physical space.
Algorithm 3 Micro-structures synthesis for theheat exchangerInput : M ( P ): a parametric micro-tile consisting of the canonical solid shape; T : a trivariate parametric deformation macro-function; H : a trivariate scalar of hot-to-cold area ratios; K : a trivariate scalar of metal thicknesses;( n x , n y , n z ): the initial dimensions of the grid enumeratingtiles M in the domain of T ; Output : MM : A micro-structure of ( n x , n y , n z ) tiles deformed tofollowing the macro-shape of T via functional composition T ( M ( P )) as influenced by the sizing from the lumped-parameter heat exchanger model; Algorithm : MM := ∅ ; for i = 1 , n x do for j = 1 , n y do for k = 1 , n z do H ijkm := hot-to-cold area ratio at the 8 cornersof the tile ( m = 1 , K ijkm := metal thicknesses at the 8 corners ofthe tile ( m = 1 , K − ijkm := T − ( K ijkm ), the deformed metalthicknesses so that after composition the cor-rect physical thicknesses are realized; P ijk := geometry/topology parameters of tile ijk , which is a function of H ijkm and K − ijkm ; M ijk := M ( P ijk ) at position ( i, j, k ); M ijk := T ( M ijk ); // via func. composition MM := MM ∪ {M ijk } ; end for end for end for return Sew( MM ); (a) (b) (c)TimeThrust Figure 7: The thrust profile (a) is provided along with the rocketfuel cross section (b), and an orthogonal (conformal) parametriza-tion. Rotating (b) defines a volume-of-revolution, shown in (c), thatspecifies the solid fuel grain. Also shown in (c) is the division of thevolume-of-revolution into layers. geometry of the solid fuel rocket casing, which defines theend state of the burn.For simplicity and to begin with, we assume that themicro-structure elements are solid and that the geometryof the solid fuel grain is a volume-of-revolution constructedby rotating a 2D planar section about the rocket’s longitu-dinal axis, not an atypical setting. See Figure 7. In otherwords, each micro-structure element is a full, possibly het-erogeneous, logical cuboid.The micro-structures-based construction scheme of theheterogeneous solid fuel grain follows Algorithm 4.Algorithm 4 builds the volume-of-revolution V ( u, v, w )(in line 1) by rotating S ( u, v ) about the rocket’s longitu-dinal axis, only to partition this trivariate (in line 2) to n layers V k ( u, v, w ), by subdividing V at n − w values. Note that each V k ( u, v, w ) is a trivariate as well -see Figure 7 (c). Each layer V k has a varying thickness.Further, the front surface (burning) area of V k varies aswell, as the burning progresses in the layer. In the limit,however, as n goes up and the thicknesses of the layersare vanishing, the front surface area can be assumed fixed.We plan the burning rate (using accelerants/retardants) soeach layer will burn completely before the next layer starts6 lgorithm 4 Heterogeneous rocket fuel optimiza-tionInput : S ( u, w ): a 2D section of rocket fuel geometry, w being a(orthogonal) parametrization of the fuel, inside out; T ( t ) , t ∈ [0 , n : micro-structure sampling rate; Output : MM : A micro-structure with heterogeneous composition AR (accelerants/retardants) of a 3D volume-of-revolutionshaped fuel grain V , with S as its cross section, satisfyingthe thrust profile T ( t ); Algorithm : V ( u, v, w ) := a volume-of-revolution trivariate definedby rotating section S ( u, w ); V k ( u, v, w ) , k = 1 , n := a partitioning in w of V ( u, v, w ) into n trivariate layers; for k = 1 , n do T otalLayerT hrust := 0; for i = 1 , n do for j = 1 , n do M ijk := Tile ij in layer V k ; A ijk := T ileF rontArea ( M ijk ); // in uv d ijk := T ileDepth ( M ijk ); // in w ; T hrust ijk := A ijk d ijk ; T otalLayerT hrust += T hrust ijk ; end for end for d k := average depth of layer V k ( u, v, w ); T hrustRatio := T ( k/n ) /T otalLayerT hrust ; for i = 1 , n do for j = 1 , n do AR ijk := T hrustRatio ∗ ( d ijk /d k ); MM := MM ∪ {M ijk ( AR ijk ) } ; end for end for end for to burn. That is, as the burning front progresses, it simul-taneously interpolates the next layer. Thus the layers serveliterally as synchronization surfaces for the timing of theburn front progression. Then, in line 7-11 of Algorithm 4,we compute for each tile M ijk in layer V k , its burn frontarea (in uv ) and depth (in w ) and estimate M ijk ’s totalthrust as its front area times its depth.Accumulating the total thrust that a layer produces as T otalLayerT hrust , we globally normalize, in line 15, therelative amount of accelerant ( AR ijk >
1) or retardant( AR ijk <
1) that is required for this layer (time step).This global normalization depends on the ratio betweenwhat is the desired thrust at this time step T ( k/n ) andthe basic thrust produced in the layer T otalLayerT hrust .Then, in line 18, the global normalization is combined witha local, tile-level, normalization. For tile M ijk , the rela-tive depth d ijk with respect to the layer’s average depthsets the local normalization. The deeper the tile, the faster it must burn (and hence requires more accelerant) to en-sure that the entire layer burns out simultaneously. Thenthe burn front interpolates the next layer synchronouslyin time.Figure 8 (a) shows one simple result, using the thrustprofile and geometry from Figure 7. Figure 8 (b) showsthat designing a fuel grain which is not a volume-of-revolution solid fuel is feasible as well. Finally, Fig-ure 8 (c), shows the potential use of porous tiles, in anaim to further increase the burning surface area and hencethe thrust. Aircraft wings have many functions and their design andpacking reflects this complexity. Wings primarily providelift (due to their outer shape), store fuel, contain mecha-nisms to provide additional lift during maneuvers such astake-off and landings, hold landing gear and the mecha-nisms to both extend and retract, and provide the abilityto steer the craft through the use of flaps. This meansthat the internals of wings contain housings for flaps, slats,landing gear, fuel tank(s), and all of the wiring and pneu-matic piping (both primary and backup) required for thefull functioning of the aircraft. The wing must be able towithstand the forces and stresses encountered during themission and therefore the internal structures that supportthe wing must be flexible, robust and light in weight.Traditionally, manufacturing has given us wing internalstructures that are a collection of a small number of spars (usually 2) and a handful of ribs that are usually orthogo-nal and are all interconnected. The spars and a couple of ribs can form a wing box that may hold fuel. Usually, the ribs have many cutouts to lighten the structure and allowfor the running of wiring and piping. Overall, the skin (atleast between the spars and the outermost rib ) is part ofthe structure (that is, the skin carries load). At times theexpanse of skin (partitioned into bays ) allows buckling (notgood) and therefore may also require stiffeners to reinforcethe structure in appropriate regions.The use of additive manufacturing in building porous,lightweight micro-structures has the potential for funda-mentally changing wing structures and therefore wing-structural design. Because the majority of internal spacein the wing remains open, routing wiring and piping doesnot degrade the structural integrity. Fuel can be storedwhile maintaining the structure by simply cordoning andwalling off parts of the micro-structural scaffolding and us-ing the segregated region to hold fuel. The wing’s skin canbe an integral part of manufacturing (and not separatelyapplied) by using tiles like those seen in Figure 3.
Stiffen-ers are not required because the unsupported expanse ofskin is now much smaller (also the skin itself, as part ofthe tile, can be locally thickened). The individual tiles canbe hollowed to further reduce the weight. See Figure 9.With the aim of exploring the basic structural responseof porous wing designs, four different configurations wereanalyzed in this section (see results in Figure 10).7a) (b) (c)
Figure 8: The thrust profile and geometry from Figure 7 yields the (sliced) result shown in (a), and using Algorithm 4. Red denotesaugmentation with accelerants, and blue, the addition of retardants. Note the high (red) burning rate at the interior middle surface, in anaim to achieve the specified high initial thrust, only to reduce into all blue (retardants) as time advances. In (b), one example of a nonvolume-of-revolution solid fuel is presented, with a thrust profile having two peaks. Finally, (c) portrays the possibility of using non-solidfuel tile (tile shown on top left in (c)), in an effort to further enhance the burning rate and hence, the thrust. In (c), the thrust is a constantcurve (while the bottom is burning faster because it is thicker). (a) (b)
Figure 9: One can model the interior of a wing, shown transparently in (a), as a trivariate (also shown transparently in (a)), in order toadaptively tile the trivariate interior with micro-structures. In (b), a more detailed structure, employing hollowed tiles is presented.
All the cases considered present the same external wingprofile and number of tiles (15 × × C continuity is imposed byenforcing the equality of the unknown variables associatedto coincident control points . In general and much likecontinuity preservation between surface patches, as long asone preserves the continuity between adjacent tiles in thedomain of the deformation function (and the deformation By enforcing the discrete elastic displacement solution to be C continuous at the interface between connected trivariates, the num-ber of unknowns is reduced to 7 million from almost 10 (51600 × × function is sufficiently continuous), the result will preservecontinuity. Finally, in the case of non-conforming tiles,the solution’s continuity could be achieved in a weak wayby means of mortaring techniques [4]. Using a multiple-core machine (70 threads were used), the computationaltime was under 40 minutes in all cases. The computationswere performed using the isogeometric analysis library iga-tools [21].In the structural analyses performed, the wing is fixedat the root, the wider transversal section that connectsthe wing with the aircraft’s fuselage (the right extrem-ity of the wing in Figure 10). We considered a linearelastic material with a Young modulus to Poisson ratioratio E/ν = 1 / . Figure 10: Structural analyses of different porous wing configurations, fixed at the root. For each case, a magnified elastic deformation ispresented (under a homogeneous lift applied to the lower wing surface), colored with von Mises stress, together with a translucent representationof the non-deformed configuration. In order to show the wing interior, a longitudinal portion of the wing has been removed from the images.In addition, below each analysis a longitudinal section of the wing is included. The four different cases considered are: a) wing design withgradual change of tile thicknesses from the root (thick) to the tip (thin); b) same configuration as a), but using thicker tiles in the interior ofthe wing, and thinner near the skin; c) wing with constant thicknesses, except at a section situated at 2 / .
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09 1 .
31d 1 .
29 2 . Table 1: Relative wing material volume (the smaller, the better)respect to the lighter design (case a); and relative wing maximumdeflection (smaller is better) respect to most flexible design (case b). wing skin, near the root, cause an increase of the stresslevel, with respect to other designs.On the other hand, the oscillatory behavior of the stressdistributions at the wing skin follows the pattern definedby the tiles distribution. In fact, the amplitude of theseoscillations is more accentuated in the case d, that presentsa thinner external skin layer.As a summary, Table 1 gathers the maximum wing de-flection, compared to the total volume of wing material(that will determine the wing self-weight, of crucial im-portance for an aircraft structure), for the four designsstudied.Looking at the table results, the heaviest design (caseb) is also the stiffest, as expected. However, as it can beseen for cases a and c, by optimizing the distribution ofthe tile’s thicknesses, it could be possible to achieve muchlighter designs without significantly reducing the wing’soverall stiffness. Further design improvements would benefit from the useof multi-objective shape optimization techniques, in whichthe maximum stress is minimized, while trying to keep alow total volume. The optimizer could operate globally,or over some neighborhood or even on individual tiles, aslong as the (geometric as well as material) continuity be-tween adjacent tiles is properly ensured. Indeed, the useof graded materials (see, e.g., [28, 17]) potentially offers awide range of possibilities in this endeavor.
Plastic profile extrusion is a manufacturing process par-ticularly suited for continuous profiles. These include pipesand floor skirtings, but also more complex geometries, suchas window panes. An extrusion line consists of three im-portant parts: (1) an extruder, responsible for melting,mixing, and transporting the raw plastic, (2) the extrusiondie, responsible for reshaping the melt to the desired pro-file, and (3) the calibration, which fixes the profile shapeduring solidification. With its high influence on the qual-ity of the final product especially in terms of shape ac-curacy, the extrusion die is certainly the component thathas been investigated in most detail. Quality criteria foran extrusion die revolve around shape accuracy of the final In a very simplified setting, considering a solid wing design whosespan and chord have a fixed length, the maximum wing deflectioncan be estimated to be proportional to the cube of its thickness (i.e.,to the cube of its volume). product: This is, in particular, influenced by the velocitydistribution at the outflow of the extrusion die, as well asthe viscoelastic stresses induced within the die. Possibledesign measures to influence these criteria are the shape ofthe flow channel within the die, and also the temperaturedistribution.In recent years, the research around extrusion dies hasbeen directed towards numerical design methods; i.e., theautomated design of the extrusion die, given a certainproduct shape as input. Ettinger gives an overview ofwork regarding shape optimization [13]. More recent re-search can be found in [24, 31, 22]. We note that all of thiswork is focused on shape optimization of the flow channel.So far, when it comes to simulations, the temperature con-trol within the extrusion die has mostly been consideredideal: In practice, this means that the process is treatedas isothermal. This assumption is justified by the heatingsystems that are currently used: The extrusion die is sim-ply wrapped with a heating band [19].It is clear that suchan approach gives no local control over the temperature.The micro-structuring approach proposed herein could- for the first time - enable local temperature control;this would be achieved without altering the overall mech-anism for heating. The extrusion die can still be wrappedwith the heating band, but the die would then be micro-structured with material of graded heat conductivity. Thiswould allow for inhomogeneous temperature distributionswithin the flow channel. As an example, confer with Fig-ure 11. We see an extrusion die intended to produce a floorskirting profile. A common problem arises when there is ahigher outflow velocity in the T-junction and a relativelylow outflow velocity at the tips of the profile, where wall-adhesion is high due to the high surface area of the wallin this regions. Mitigation of these effects based on flow-channel shape has been investigated in [27]. A similareffect could be achieved with locally reducing tempera-ture in the T-junction, while, at the same time, increasingtemperature in the tips. Figure 11 illustrates a possibledistribution of micro-structures that would result in theaforementioned temperature distribution.Furthermore, the use of micro-structuring opens up therealm of developing systems using entirely new tempera-ture control approaches. For example, one could use vari-able inductive heating, whereby the micro-structures aregraded with respect to electrical conductivity.For this application, the microstructures will be parame-terized in terms of their geometry and their material prop-erties. These can then be modified locally with respect tothe design objective of homogeneous velocity distribution[12]. In order to keep the number of optimization param-eters manageable, expert knowledge will be included intothe parameterization. We envision the use of a trust-regionoptimization algorithm like BOBYQA [23].10a) (b)
Figure 11: Local control over heating in an extruder (in magenta). Red tiles are insulators (for both heat and electricity). Yellow tiles areonly thermally conductive, while green tiles are electrically (and thermally) conductive. Based on the diameters of the cross sections of thegreen tiles, the electrical resistance can be controlled, which locally affects the amount of heat generated when electric current flows throughthe green tiles. (a) shows a view from the outlet, while (b) shows a close-up side view.
5. Conclusions and Future Work
We have presented a design paradigm exploiting para-metric tiling in micro-structures. Control over both theshape and the materials were presented. Further degreesof freedom to optimize include • Control over the size of the micro-structure grid orrecursive level or embedding nano-structures withinmicro-structures, etc. [17]. • Control over the topology of the tiles, employing tileswith different topologies in different locations. Onesuch potential example is shown in Figure 12 wherebifurcations are employed. This example employs tileswith (trimmed) surfaces and contiguous surfaces inthe tiles that are not conforming. For simpler analysisand optimization, it is better to design with conform-ing (preferably untrimmed) trivariate based tiles. • Control over the macro-shape or the deformationfunction. One example for such an ability is shown inFigure 13 that synthesizes longer fingers to the heatsink. Compare with Figure 2.The example in Figure 12 demonstrates the potentialin an effort to keep all tiles of uniform size. Such a con-straint is typically application driven (i.e. minimal thick-ness walls in additive manufacturing) and is not a limitof the presented process. The micro-tiles can be deformedarbitrarily. Further, as many tiles as desired can fit intoone deformation macro-function. It is limited only by com-puter memory needs and computational costs.Up to continuity requirments, the different tiles in thedomain of the deformation macro-function are indepen-dent and can be arbitrarily different (and even random).Each tile can present a different topology, geometry ormaterial properties. How this generality will be fully ex-ploited in design is yet to be seen.There has been a great deal of recent interest in the useof Topological Optimization to generate interesting (andsometimes unintuitive) structural designs. This focus is partially due to the commonalities with additive manu-facturing that both use the same volumetric underpin-ning (voxels). This means that a structural design canbe simply printed without translation. But like any en-gineering tool there are limitations, which in this case in-clude: deep optimal designs (which can be fragile), diffi-culties when surface smoothness is critical, compatibilitywith contemporary CAD systems, and dealing with de-signs where (structural) analysis is only one of many dis-ciplines in play.It is the last point above (designs for multi-physics de-vices) when micro-structures provide a general viable al-ternative to voxels and therefore the possibility to designin multidisciplinary settings. Examples of this can be seenthroughout Section 4 where the outer/macro shape neednot be rectilinear and the micro-shapes have few limita-tions. The difficulty in this design setting is that the num-ber of parameters that drive the design through optimiza-tion can be quite large (and some continuous). This can beeffectively handled in a gradient-based optimization man-ner where the parametric derivatives for the entire problemare available. Generating these derivatives can efficientlybe accomplished through the use of tightly coupled physicssolvers that include their Adjoint so that the full Jacobianof the coupled problem can be made available. Then thechain-rule can be applied to couple the Jacobian to theparametric derivatives produced from differentiating thegeometry construction.
Acknowledgments
This research was supported in part by the ISRAELSCIENCE FOUNDATION (grant No. 597/18) and inpart with funding from the Defense Advanced ResearchProjects Agency (DARPA), under contract HR0011-17-2-0028. The views, opinions and/or findings expressed arethose of the author and should not be interpreted as rep-resenting the official views or policies of the Departmentof Defense or the U.S. Government. Pablo Antolin andAnnalisa Buffa gratefully acknowledge the support of the11a)(b) (c)
Figure 12: Two views on a B-spline surface micro-structure in the shape of a delta wing with macro-tiles’ surfaces that employs bifurcationsto change topology. The surfaces shown on the right view (a) are automatically merged by using bifurcations in an effort to bound theminimal/maximal tile size. Starting from four rows of tiles, near the root of the wing, it goes into two rows one third of the way along thewing, and then into one row toward the tip of the wing. Similar shrinkage in the number of tiles can also be observed from above (on theleft in (b)). (c) shows the a-priori defined parameteric tiles with the different topologies. The right-most tile consists of trimmed surfaceswhereas the other three tiles of tensor product surfaces only.Figure 13: Another degree of freedom that can be exploited is themacro shape of the micro-structure. Here, the amount of twisting ofthe ring is increased compared to Figure 2.
European Research Council, through the ERC AdG n.694515 - CHANGE.
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