Research Papers: Design of Mechanisms and Robotic Systems

Computational Synthesis of Large Deformation Compliant Mechanisms Undergoing Self and Mutual Contact

[+] Author and Article Information
Prabhat Kumar

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: prabhatkumar.rns@gmail.com

Anupam Saxena

Mechanical Engineering,
Indian Institute of Technology Kanpur,
Kanpur 208016, India
e-mail: anupams@iitk.ac.in

Roger A. Sauer

Aachen Institute for Advanced Study in
Computational Engineering Science (AICES),
RWTH Aachen University,
Templergraben 55,
52056 Aachen, Germany

1Corresponding author.

2Present Address: Department of Precision and Microsystems Engineering, Faculty of 3mE, Delft University of Technology, Mekelweg 2, Delft 2628 CD, The Netherlands.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 28, 2018; final manuscript received July 29, 2018; published online October 10, 2018. Assoc. Editor: Nam H. Kim.

J. Mech. Des 141(1), 012302 (Oct 10, 2018) (13 pages) Paper No: MD-18-1399; doi: 10.1115/1.4041054 History: Received May 28, 2018; Revised July 29, 2018

Topologies of large deformation contact-aided compliant mechanisms (CCMs), with self and mutual contact, exemplified via path generation applications, are designed using the continuum synthesis approach. Design domain is parameterized using honeycomb tessellation. Assignment of material to each cell, and generation of rigid contact surfaces, are accomplished via suitably sizing and positioning negative circular masks using the stochastic hill-climber search. To facilitate contact analysis, boundary smoothing is implemented. Mean value coordinates are employed to compute shape functions, as many regular hexagonal cells get degenerated into irregular, concave polygons as a consequence of boundary smoothing. Both geometric and material nonlinearities are considered. The augmented Lagrange multiplier method with a formulated active set strategy is employed to incorporate both self and mutual contact. Synthesized contact-aided compliant continua trace paths with single, and importantly, multiple kinks and experience multiple contact interactions pertaining to both self and mutual contact modes.

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Mankame, N. D. , and Ananthasuresh, G. , 2004, “ Topology Optimization for Synthesis of Contact-Aided Compliant Mechanisms Using Regularized Contact Modeling,” Comput. Struct., 82(15–16), pp. 1267–1290. [CrossRef]
Kumar, P. , Sauer, R. A. , and Saxena, A. , 2016, “ Synthesis of c0 Path-Generating Contact-Aided Compliant Mechanisms Using the Material Mask Overlay Method,” ASME J. Mech. Des., 138(6), p. 062301. [CrossRef]
Wriggers, P. , 2006, Computational Contact Mechanics, Springer, Heidelberg, Germany.
Howell, L. L. , 2001, Compliant Mechanisms, Wiley, New York.
Ananthasuresh, G. , Kota, S. , and Gianchandani, Y. , 1994, “ A Methodical Approach to the Design of Compliant Micromechanisms,” Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, pp. 189–192.
Nishiwaki, S. , Frecker, M. I. , Min, S. , and Kikuchi, N. , 1998, “ Topology Optimization of Compliant Mechanisms Using the Homogenization Method,” Int. J. Numer. Methods Eng., 42(3), pp. 535–559. [CrossRef]
Frecker, M. , Ananthasuresh, G. , Nishiwaki, S. , Kikuchi, N. , and Kota, S. , 1997, “ Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization,” ASME J. Mech. Des., 119(2), pp. 238–245. [CrossRef]
Saxena, A. , and Ananthasuresh, G. , 2000, “ On an Optimal Property of Compliant Topologies,” Struct. Multidiscip. Optim., 19(1), pp. 36–49. [CrossRef]
Sigmund, O. , 1997, “ On the Design of Compliant Mechanisms Using Topology Optimization,” J. Struct. Mech., 25(4), pp. 493–524.
Saxena, A. , and Ananthasuresh, G. , 2001, “ Topology Synthesis of Compliant Mechanisms for Nonlinear Force-Deflection and Curved Path Specifications,” ASME J. Mech. Des., 123(1), pp. 33–42. [CrossRef]
Pedersen, C. B. , Buhl, T. , and Sigmund, O. , 2001, “ Topology Synthesis of Large-Displacement Compliant Mechanisms,” Int. J. Numer. Methods Eng., 50(12), pp. 2683–2705. [CrossRef]
Saxena, A. , 2005, “ Synthesis of Compliant Mechanisms for Path Generation Using Genetic Algorithm,” ASME J. Mech. Des., 127(4), pp. 745–752. [CrossRef]
Swan, C. C. , and Rahmatalla, S. F. , 2004, “ Design and Control of Path-Following Compliant Mechanisms,” ASME Paper No. DETC2004-57441.
Ullah, I. , and Kota, S. , 1997, “ Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptors and Global Search Methods,” ASME J. Mech. Des., 119(4), pp. 504–510. [CrossRef]
Zahn, C. T. , and Roskies, R. Z. , 1972, “ Fourier Descriptors for Plane Closed Curves,” IEEE Trans. Comput., 100(3), pp. 269–281. [CrossRef]
Rai, A. K. , Saxena, A. , and Mankame, N. D. , 2007, “ Synthesis of Path Generating Compliant Mechanisms Using Initially Curved Frame Elements,” ASME J. Mech. Des., 129(10), pp. 1056–1063. [CrossRef]
Rai, A. K. , Saxena, A. , and Mankame, N. D. , 2010, “ Unified Synthesis of Compact Planar Path-Generating Linkages With Rigid and Deformable Members,” Struct. Multidiscip. Optim., 41(6), pp. 863–879. [CrossRef]
Saxena, A. , 2008, “ A Material-Mask Overlay Strategy for Continuum Topology Optimization of Compliant Mechanisms Using Honeycomb Discretization,” ASME J. Mech. Des., 130(8), p. 082304. [CrossRef]
Saxena, A. , 2011, “ An Adaptive Material Mask Overlay Method: Modifications and Investigations on Binary, Well Connected Robust Compliant Continua,” ASME J. Mech. Des., 133(4), p. 041004. [CrossRef]
Saxena, A. , 2011, “ Topology Design With Negative Masks Using Gradient Search,” Struct. Multidiscip. Optim., 44(5), pp. 629–649. [CrossRef]
Saxena, A. , and Sauer, R. A. , 2013, “ Combined Gradient-Stochastic Optimization With Negative Circular Masks for Large Deformation Topologies,” Int. J. Numer. Methods Eng., 93(6), pp. 635–663. [CrossRef]
Mankame, N. D. , and Ananthasuresh, G. , 2002, “ Contact Aided Compliant Mechanisms: Concept and Preliminaries,” ASME Paper No. DETC2002/MECH-34211.
Mankame, N. , and Ananthasuresh, G. , 2007, “ Synthesis of Contact-Aided Compliant Mechanisms for Non-Smooth Path Generation,” Int. J. Numer. Methods Eng., 69(12), pp. 2564–2605. [CrossRef]
Reddy, B. V. S. N. , Naik, S. V. , and Saxena, A. , 2012, “ Systematic Synthesis of Large Displacement Contact-Aided Monolithic Compliant Mechanisms,” ASME J. Mech. Des., 134(1), p. 011007. [CrossRef]
Tummala, Y. , Wissa, A. , Frecker, M. , and Hubbard, J. E. , 2014, “ Design and Optimization of a Contact-Aided Compliant Mechanism for Passive Bending,” ASME J. Mech. Rob., 6(3), p. 031013. [CrossRef]
Kumar, P. , Saxena, A. , and Sauer, R. A. , 2017, “ Implementation of Self Contact in Path Generating Compliant Mechanisms,” Microactuators and Micromechanisms, Springer, Cham, pp. 251–261.
Cannon, J. R. , and Howell, L. L. , 2005, “ A Compliant Contact-Aided Revolute Joint,” Mech. Mach. Theory, 40(11), pp. 1273–1293. [CrossRef]
Moon, Y.-M. , 2007, “ Bio-Mimetic Design of Finger Mechanism With Contact Aided Compliant Mechanism,” Mech. Mach. Theory, 42(5), pp. 600–611. [CrossRef]
Aguirre, M. , Hayes, G. , Frecker, M. , Adair, J. , and Antolino, N. , “ Fabrication and Design of a Nanoparticulate Enabled Micro Forceps,” ASME Paper No. DETC2008-49917.
Mehta, V. , Frecker, M. , and Lesieutre, G. A. , 2009, “ Stress Relief in Contact-Aided Compliant Cellular Mechanisms,” ASME J. Mech. Des., 131(9), p. 091009. [CrossRef]
Saxena, A. , 2013, “ A Contact-Aided Compliant Displacement-Delimited Gripper Manipulator,” ASME J. Mech. Rob., 5(4), p. 041005. [CrossRef]
Calogero, J. , Frecker, M. , Hasnain, Z. , and Hubbard, J. E., Jr , 2016, “ A Dynamic Spar Numerical Model for Passive Shape Change,” Smart Mater. Struct., 25(10), p. 104006. [CrossRef]
Sauer, R. A. , and De Lorenzis, L. , 2015, “ An Unbiased Computational Contact Formulation for 3d Friction,” Int. J. Numer. Methods Eng., 101(4), pp. 251–280. [CrossRef]
Saxena, R. , and Saxena, A. , “ On Honeycomb Parameterization for Topology Optimization of Compliant Mechanisms,” ASME Paper No. DETC2003/DAC-48806.
Langelaar, M. , 2007, “ The Use of Convex Uniform Honeycomb Tessellations in Structural Topology Optimization,” Seventh World Congress on Structural and Multidisciplinary Optimization, Seoul, South Korea, May 21–25, pp. 21–25.
Saxena, R. , and Saxena, A. , 2007, “ On Honeycomb Representation and Sigmoid Material Assignment in Optimal Topology Synthesis of Compliant Mechanisms,” Finite Elem. Anal. Des., 43(14), pp. 1082–1098. [CrossRef]
Talischi, C. , Paulino, G. H. , and Le, C. H. , 2009, “ Honeycomb Wachspress Finite Elements for Structural Topology Optimization,” Struct. Multidiscip. Optim., 37(6), pp. 569–583. [CrossRef]
Talischi, C. , Paulino, G. H. , Pereira, A. , and Menezes, I. F. , 2012, “ Polytop: A Matlab Implementation of a General Topology Optimization Framework Using Unstructured Polygonal Finite Element Meshes,” Struct. Multidiscip. Optim., 45(3), pp. 329–357. [CrossRef]
Saxena, A. , 2010, “ On an Adaptive Mask Overlay Topology Synthesis Method,” ASME Paper No. DETC2010-29113.
Talischi, C. , Paulino, G. H. , Pereira, A. , and Menezes, I. F. , 2012, “ Polymesher: A General-Purpose Mesh Generator for Polygonal Elements Written in Matlab,” Struct. Multidiscip. Optim., 45(3), pp. 309–328. [CrossRef]
Talischi, C. , Paulino, G. H. , Pereira, A. , and Menezes, I. F. , 2010, “ Polygonal Finite Elements for Topology Optimization: A Unifying Paradigm,” Int. J. Numer. Methods Eng., 82(6), pp. 671–698.
Kumar, P. , and Saxena, A. , 2015, “ On Topology Optimization With Embedded Boundary Resolution and Smoothing,” Struct. Multidiscip. Optim., 52(6), pp. 1135–1159. [CrossRef]
Corbett, C. J. , and Sauer, R. A. , 2014, “ Nurbs-Enriched Contact Finite Elements,” Comput. Methods Appl. Mech. Eng., 275, pp. 55–75. [CrossRef]
Kumar, P. , and Saxena, A. , 2013, “ On Embedded Recursive Boundary Smoothing in Topology Optimization With Polygonal Mesh and Negative Masks,” First International and 16th National Conference on Machines and Mechanisms (iNaCoMM2013), Roorkee, India, Dec. 18–20, pp. 568–575.
Floater, M. S. , 2003, “ Mean Value Coordinates,” Comput. Aided Geom. Des., 20(1), pp. 19–27. [CrossRef]
Hormann, K. , and Floater, M. S. , 2006, “ Mean Value Coordinates for Arbitrary Planar Polygons,” ACM Trans. Graph., 25(4), pp. 1424–1441. [CrossRef]
Sukumar, N. , and Tabarraei, A. , 2004, “ Conforming Polygonal Finite Elements,” Int. J. Numer. Methods Eng., 61(12), pp. 2045–2066. [CrossRef]
Sukumar, N. , and Malsch, E. , 2006, “ Recent Advances in the Construction of Polygonal Finite Element Interpolants,” Arch. Comput. Methods Eng., 13(1), pp. 129–163. [CrossRef]
Zienkiewicz, O. C. , and Taylor, R. L. , 2005, The Finite Element Method for Solid and Structural Mechanics, Butterworth-Heinemann, Oxford, UK.
Wriggers, P. , 2008, Nonlinear Finite Element Methods, Springer Science & Business Media, Berlin, Germany.
Kumar, P. , 2017, “ Synthesis of Large Deformable Contact-Aided Compliant Mechanisms Using Hexagonal Cells and Negative Circular Masks,” Ph.D. thesis, Indian Institute of Technology Kanpur, Kanpur, India.
Knuth, D. E. , 1998, The Art of Computer Programming: Sorting and Searching, Vol. 3, Pearson Education, Upper Saddle River, NJ.
Russell, S. J. , and Norvig, P. , 2003, Artificial Intelligence: A Modern Approach, 2nd ed., Pearson Education, Upper Saddle River, NJ.
Tikhonov, A. N. , Goncharsky, A. , Stepanov, V. , and Yagola, A. G. , 2013, Numerical Methods for the Solution of Ill-Posed Problems, Vol. 328, Springer Science & Business Media, Dordrecht, The Netherlands.
Kumar, P. , Sauer, R. A. , and Saxena, A. , 2015, “ On Synthesis of Contact Aided Compliant Mechanisms Using the Material Mask Overlay Method,” ASME Paper No. DETC2015-47064.
Guest, J. K. , Prévost, J. , and Belytschko, T. , 2004, “ Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions,” Int. J. Numer. Methods Eng., 61(2), pp. 238–254. [CrossRef]
Canfield, S. L. , Chlarson, D. L. , Shibakov, A. , Richardson, J. D. , and Saxena, A. , 2007, “ Multi-Objective Optimization of Compliant Mechanisms Including Failure Theories,” ASME Paper No. DETC2007-35618.


Grahic Jump Location
Fig. 6

Design specification and specified paths for four different CCMs: (a) design specification for all examples with the initial guess for mask parameters, and (b) desired output paths for all four CCMs are depicted

Grahic Jump Location
Fig. 5

A schematic flowchart for a single design iteration within the proposed optimization approach

Grahic Jump Location
Fig. 4

Schematic describing self-contact when two surface regions Γs1 and Γs2 of the same body penetrate each other. The hatched portion shows that the top part of the body has penetrated into the bottom part. Arrows are marked on the boundary to indicate orientation and to differentiate between two interacting surfaces. For point xs∈Γs1, point xp∈Γs2 is the nearest neighbor. xp1∈Γs1 or xp2∈Γs1 are also candidate nearest neighbors since normal gaps g1·np1 and g2·np2 are negative in both cases. Such candidate nearest neighbors are discarded noting that the dot products between normals, e.g., nps·np1 and nps·np2 are positive.

Grahic Jump Location
Fig. 3

A schematic diagram for the contact formulation: (a) deformed configurations, (b) contact description with tractions,(c) closest point evaluation, and (d) contact discretization, piecewise linear segments approximate interacting boundaries

Grahic Jump Location
Fig. 2

V-notches furnish jumps in boundary normals (Γcm) (a) which are subdued via boundary smoothing (c) to facilitate contact analysis. (b) depicts the way boundary smoothing is performed. (d) shows that for higher values of β, some elements may experience flipping. (a) Body Bm without boundary smoothing, (b) boundary smoothing scheme, (c) body Bs with boundary smoothing; considering β = 1, and (d) higher values of β can result in significant distortion of elements, and also, element flipping.

Grahic Jump Location
Fig. 1

A set of hexagonal cells ΩH discretize the design space Ω. Negative circular masks ΩM (circles) superposed on Ω help determine the material state of ΩH. Five variables (xp, yp, rp, sp, fp) define each mask. (xp, yp) and rp are center coordinates and the radius of the pth mask. sp = 1 (b) implies a rigid contact surface (dark, filled circular regions) of radius fprp is generated within the pth mask while with sp = 0 (a), no contact surface is generated. ρH) = 0 (ΩH⊂ any ΩM) implies a void material state while ρH) = 1 indicates full material state. Rigid contact surfaces can interact with the continuum. In addition, surfaces (e.g., Γs1 and Γs2) of the continuum may interact in self-contact mode.

Grahic Jump Location
Fig. 7

Final solutions for CCM I–IV: left column: final continua with positions of masks and mutual contact surfaces shown, middle and right columns: comparison of paths generated and final deformed configurations. Active contact regions are depicted within dashed circles.

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Fig. 8

Deformed configurations of CCMs at stages A, B and C (Fig. 7): (a) CCM I, (b) CCM II, (c) CCM III, and (d) CCM IV

Grahic Jump Location
Fig. 9

(a) Prototypes of CCMs I–IV in their undeformed (left) and deformed (middle and right) configurations (b) Paths from the prototypes (PP) compared with those traced by the CCMs (AP) in simulation. Left-right: CCMs I–IV.

Grahic Jump Location
Fig. 12

Computational cost and convergence history with INTEL CORE(TM) i5-6000 CPU at 2.70 GHz. Solution (a) is obtained after 6000 function evaluations. A feasible candidate continuum, that having the required input/output ports, some fixed boundaries, and that is well connected, is not available for the first 880 iterations. The overall synthesis takes about 12 h. (a) CCM after 6000 iterations and (b) convergence history.

Grahic Jump Location
Fig. 11

Paths and CPU time with various Gauss points, Key: GP–Gauss points, CT(s)–CPU time in seconds

Grahic Jump Location
Fig. 10

Paths traced by CCMs I–IV meshed with different number of hexagonal cells (inset). (d) desired path is rotated for better comparison. SP represents the desired path. (a) CCM I, (b) CCM II, (c) CCM III, and (d) CCM IV.



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