Lattice Structure Design with Linear Optimization in the Field of Generative Design

Transkript

1 Lattice Structure Design with Linear Optimization in the Field of Generative Design

2 Structure 1. Motivation 2. Workflow 3. Linear Statics for Frame Member 4. Optimization Model 5. CAD-Integration 6. Results 2

3 TOR-Pyramid : Generative Design Optimierungsperspektive Entscheiden 1 Ingenieursperspektive 1 2 Was ist die Funktion? Was ist mein Ziel? 2? 1 2 Optimierungsschwerpunkt? Konstruktionsziele? 3 Wie groß ist das Spielfeld? 3 3 Wie groß ist der Bauraum? 4 5 Suche das optimale System! Evaluiere mit Modellen! Handeln Strukturoptimierung! Evaluiere mit ingenieurtechnischem Erfahrungswissen 6 Validiere! 6 6 Validierungsrechnung (Numerik) 7 Realisiere! 7 7 Additive Fertigungsverfahren 3

4 Generative Design leads to a radical Rethinking in the Area of Construction human input: case of loading assembly space iterative process linear optimization CAD / CAE Quelle: in Anlehnung an Creo Parametric Using Lattices in Creo Simulate 4

5 Motivation - Generative Design human input: case of loading assembly space computed optimal solutions optimization 5

6 Motivation - Classical Apporach FEA FEA needs much computational power Linear optimization should be able to show better scalability Quelle: in Anlehnung an L.K. Engineering Design of spacecraft components for Additive manufacturing 6

7 Bridging Linear Optimization and Additive Manufacturing Determine assembly space Determine load case Linear optimization Construction proposal Finite element analysis Adjustment of possible restrictions No Target requirements met? Additive Manufacturing.gcode.STL.CATpart Yes 8

8 Recap statics Allgemeines Gleichgewichtsaxiom Ein mechanisches System befindet sich im Zustand der Ruhe (Gleichgewicht) oder im Zustand der gleichförmigen geradlinigen Bewegung, wenn die angreifenden Kräfte und Momente in ihrer Wirkung auf einen starren Körper einer verschwindenden Dyname mit R = 0 und M p = 0 äquivalent sind. n R = F i = 0 i=1 F ix = 0 F iy = 0 F iz = 0 n m M p = M (p) = r i F i + M i = 0 i=1 i=1 σ(+r iy F iz - r iz F iy ) + σ M ix = 0 σ( r ix F iz + r iz F ix ) + σ M iy = 0 σ(+r ix F iy - r iy F ix ) + σ M iz = 0 9

9 Recap graph theory graph theory or in particular a flow network appears suitable to optimize in terms of uniform force distribution v V s, t : f(e) = f(e) f = f e f(e) δ (v) {e = x, v E x V } e δ{ } v e δ + v e {δ + } v e δ v δ + (v) {e = x, v E x V } 11

10 Bridging Linear Optimization and Additive Manufacturing Determine assembly space Determine load case Linear optimization Construction proposal Finite element analysis Adjustment of possible restrictions No Target requirements met? Additive Manufacturing.gcode.STL.CATpart Yes 12

11 Determine Assembly Space 2D 13

12 Simplification from 3D to 2D 26 possible bars/restrictions 8 possible bars/restrictions 14

13 Determine Assembly Space 2D case 1 horizontal i Knoten i = 1 case 1 vertical i Knoten i = 1 k NQ QLoc k 1) case 3 horizontal i Knoten i mod m = 0 i 2m i n m m case 3 vertical i noten i mod m = 0 i 2m i n m m case 7 horizontal i Knoten i 2 i m 1 case 7 vertical i Knoten i 2 i m 1 ( k NQ QLoc k i) 15

14 Determine Assembly Space from 2D to 3D 2D 9 cases 8 possible bars per node with 2 equations 144 possible restrictions to consider 3D 27 cases 26 possible bars per node with 3 equations 2106 possible restrictions to consider 16

15 Determine Load Case

16 Bridging Linear Optimization and Additive Manufacturing Determine assembly space Determine load case Linear optimization Construction proposal Finite element analysis Adjustment of possible restrictions No Target requirements met? Additive Manufacturing.gcode.STL.CATpart Yes 18

17 Optimization Model F v i m 1,i F h i m 1,i i 1,i i,i 1 F h F h i,i m F i m+1,i i m 1,i v i,i m+1 F F v F v v i-m-1 i-m i-m+1 i i-1 i m,i F v i m+1,i F h i,i m 1 i,i m+1 F F h h i+1 i,i+1 i+1,i F h F h F h i,i+m 1 i+m-1 i+m i+m+1 F h i,i+m+1 F v i+m 1,i F v i,i+m 1 F v i,i+m F v i,i+m+1 F v i+m+1,i F h i+m 1,i F v i+m,i F h i+m+1,i 19

18 Reduction from 26 to 18 restrictions due to orthogonal force distripution F ix = 0 F iy = 0 F iz = 0 21

19 Simulation of a restricted node 22

20 Recap statics Allgemeines Gleichgewichtsaxiom Ein mechanisches System befindet sich im Zustand der Ruhe (Gleichgewicht) oder im Zustand der gleichförmigen geradlinigen Bewegung, wenn die angreifenden Kräfte und Momente in ihrer Wirkung auf einen starren Körper einer verschwindenden Dyname mit R = 0 und M p = 0 äquivalent sind. n R = F i = 0 i=1 F ix = 0 F iy = 0 F iz = 0 n m M p = M (p) = r i F i + M i = 0 i=1 i=1 σ(+r iy F iz - r iz F iy ) + σ M ix = 0 σ( r ix F iz + r iz F ix ) + σ M iy = 0 σ(+r ix F iy - r iy F ix ) + σ M iz = 0 23

21 Equilibrium of forces of a node Modeling task n R = F i = 0 i=1 Kraftkomponente_X {i in Knoten: i==(x*z)-x+1} [ ] : +F(i,i+1)+a*F(i,i-x+1)+a*F(i,i+(x*z)+1)+b*F(i,i+(x*z)-x+1)+Q[i] [ ] = 0; Kraftkomponente_Y {i in Knoten: i==(x*z)-x+1} [ ] : +a*f(i,i+(x*z)+1)+b*f(i,i+(x*z)-x+1)+f(i,i+(x*z))+a*f(i,i+(x*z)-x)+q[i] [ ] = 0; Kraftkomponente_Z {i in Knoten: i==(x*z)-x+1} [ ] : -a*f(i,i-x+1)-b*f(i,i+(x*z)-x+1)-a*f(i,i+(x*z)-x)-f(i,i-x)+q[i]+bz[i] [ ] = 0; position in assembly space equilibrium condition in all spatial directions 24

22 Recap statics Allgemeines Gleichgewichtsaxiom Ein mechanisches System befindet sich im Zustand der Ruhe (Gleichgewicht) oder im Zustand der gleichförmigen geradlinigen Bewegung, wenn die angreifenden Kräfte und Momente in ihrer Wirkung auf einen starren Körper einer verschwindenden Dyname mit R = 0 und M p = 0 äquivalent sind. n R = F i = 0 i=1 F ix = 0 F iy = 0 F iz = 0 n m M p = M (p) = r i F i + M i = 0 i=1 i=1 σ(+r iy F iz - r iz F iy ) + σ M ix = 0 σ( r ix F iz + r iz F ix ) + σ M iy = 0 σ(+r ix F iy - r iy F ix ) + σ M iz = 0 25

23 Equilibrium of moments n m M p = M (p) = r i F i + M i = 0 i=1 i=1 Modeling task {i in Knoten} : Ly[i] = floor(i-1/x*z); {i in Knoten} : Lx[i] = (i mod x*z-1) mod x; planes within assembly space external force lever arm bearing force lever arm {b in NB} : sum{i in Knoten: Q[i]!= 0} Q[i]*(Ly[i]-Ly[BLoc[b]])+sum{c in NB} Bz[c]* (Ly[BLoc[b]]-Ly[BLoc[c]]) = 0; {b in NB} : sum{i in Knoten: Q[i]!= 0} Q[i]*(Lx[i]-Lx[BLoc[b]])+sum{c in NB} Bz[c]* (Lx[BLoc[b]]-Lx[BLoc[[c]]) = 0; 26

24 Bridging Linear Optimization and Additive Manufacturing Determine assembly space Determine load case Linear optimization Construction proposal Finite element analysis Adjustment of possible restrictions No Target requirements met? Additive Manufacturing.gcode.STL.CATpart Yes 27

25 Illustration of the construction space as a node network EOSINT P x 380 x 580 mm Assembly space for LP Quelle: in Anlehnung an EOS : AM Basic Design Rules for Additive Manufacutring 28

26 Illustration of the construction space as a node network 29

27 Automated generation of frameworks in CAD Consistent and valid CAD data from scratch Material data and standard 30

28 Automated generation of frameworks in CAD data from linear optimization CAD data 31

29 CAD example instance 32

30 CAD example instance 33

31 CAD example instance 34

32 Automated generation of frameworks in CAD 35

33 Many thanks for your attention! Christian Reintjes 37

34 References Creo Parametric. Online available at %2Fmodstr%2Ffeatures%2Freference%2Fsimulate_lattices.html%23, checked on L.K.Engineering - Design of spacecraft components for Additive manufacturing. checked on EOS Additive Manufacturing (AM) : Basic Design Rules for Additive Manufacturing 8b/EOS-Basic-Design-Rules_Additive-Manufacturing_EN.pdf, checked on Christian Reintjes 38