EU Regional School - Saxena Seminar
Prof. Dr. Saxena - Systematic Synthesis of Large Displacement Compliant Mechanisms: A Structural Optimization Approach
Indian Institute of Technology Kanpur, India
Topology optimization entails determining optimal material layout of a continuum within a specified design domain for a desired set of objectives. Irrespective of the parameterization used, material state of a point/sub-region/cell/finite-element should ideally toggle between ‘solid’ and ‘void’ states eventually leading to a well-defined optimized solution. In other words, material assignment must, ideally, be discrete.
Two parameterization schemes will be described – line element and honeycomb tessellation, both in context of synthesizing large displacement compliant continua. The latter could be monolithic (single-piece), partially compliant, or some of their members could physically interact via ‘contact.’ With line element parameterization, topology-size decoupling will be emphasized as in how it helps pose topology, shape and size optimization independent of each other. Furthermore, such a framework also helps in introducing, say, rigid members and pin joints within a network of flexible frames leading to the possibility of synthesizing partially compliant continua. As discrete material assignment is strictly adhered to, notwithstanding efficiency, the solitary choice of using a stochastic optimization approach also helps in rejecting ‘non-convergent’ (from the perspective of large displacement analysis) intermediate continua, which, otherwise, tends to impede the functioning of a gradient-based optimization algorithm. Co-rotational beam theory to model frames undergoing geometrically large displacements will be briefed followed by a Fourier Shape Descriptors based objective and a random mutation hill climber algorithm to synthesize ‘path-generating’ continua exemplifying large displacement compliant mechanisms.
In continuum parameterization, traditionally, each sub-region is represented by a single (or a set of) Lagrangian (e.g., triangular/rectangular) type finite element(s). With such parameterization however, numerous connectivity singularities such as checkerboards, point flexures, layering/islanding, right-angled notches and ‘blurred’ boundaries are observed, unless ‘additional’ filtering-type methods are used. Use of honeycomb tessellation will be described. As hexagonal cells provide edge-connectivity between any two contiguous cells, most geometric singularities get eliminated naturally. However, numerous ‘V’ notches persist at continuum boundaries which are subdued via a winged-edge data structure based boundary resolution scheme. Consequently, many hexagonal cells get morphed into concave cells. Finite element modeling of each cell is therefore accomplished using the Mean-Value Coordinate based shape functions that can cater to any generic polygonal shape. Overlaying negative circular masks are used to assign material states to sub-regions. Their radii and center coordinates are varied in a manner that material is removed from sub-regions lying beneath the masks so that remnant, unexposed sub-regions constitute a realizable continuum.
Honeycomb tessellation, boundary smoothing and Mean Value Coordinates based analysis, all pave way to synthesize Contact-aided Compliant Mechanisms (CCMs) with suitable modifications in the topology optimization formulation which will be highlighted. Augmented Lagrangian method along with active set constraints has been used for contact analysis. Synthesis of large displacement CCMs is exemplified via path generation. Self-contact between continuum subregions undergoing large deformation can also occur, a feature that could be used by such continua to assist them in performing special tasks, such as, attaining negative stiffness and static balancing. Contact analysis is extended to cater to deforming bodies. Numerous examples will be presented to showcase a variety of design features and highlight the ability of Contact-aided Compliant Mechanisms to achieve/accomplish complex kinematic tasks.
Lecture Material I
Lecture Material II