Appendix
ASSUMPTIONS
Structure:
- Simple beams & plates in quadrilateral shapes
- Dimensions: max 600mm, min 100mm (length), max 10mm, min 0.1mm (depth)
- Horizontal laminated layers without curvature
- Perfect lamination & bonding between adjacent layers for cohesive response
- Supports for beams consist of edge and intermediate supports (rollers, hinges, clamped) with no partial supports
- Supports for plates consist only of edge supports (rollers, hinges, clamped) with no partial supports
- Aspect ratios of L/W=[1,2] & L/d=[10,200]
- Ply angles variation = [0,15,30,45,90]
- Each laminated layer is of homogeneous, isotropic material
FEM model:
- Linear strain model used with linear strain, isoparametric quadratic quadrilateral element with 8 nodes
- Linear constitutive equations are used in formulation, strain-displacement relations are based on first order shear theory and electric potential functions have linear variations across the thickness of the piezoelectric actuator layers
- 5 degrees of freedom within each node, imposed by FEM modelling, corresponding to [u,v,w,q x,q y]
- Loads corresponding to d.o.f. are 3 translational loads and 2 rotational loads
- Magnitudes of the loads are determined empirically through trial simulations to achieve realistic modelling for both beams and plates
- The types of loads investigated are concentrated point loads and uniformly distributed loads in different directions. The FYP program suit also includes other load configurations for future investigations & extensions
- FEM model neglects non-linear characteristics (thus unable to affirm & expand on the works of Ajit(10) )
- The LSQ elements of the FEM are programmed to be fixed in shape thus mesh density throughout modelled structure is uniform with no local mesh concentration allowed
- The mesh of LSQ elements also assumes uniform element depth, thus structure must be of uniform thickness throughout its plan
- Equilibrium and compatibility of uncontrolled and controlled structural responses are imposed on all nodes only and may not be within elements, thus the mesh density reflects the resolution of smart control response modelling
Smart control:
- Piezoceramic actuators are used only for static control, without sensors
- The actuators are surface bonded perfectly to the structure to be controlled
- The actuators are in pairs (double-sided), with single-sided actuators either on top or bottom surfaces to be for future investigations
- The actuator pairs operate in out-of -phase without in-plane forces to produce pure bending & curvature at the placement and voltages applied (in-phase operation to be for future investigations that induces in-plane forces)
- Localised distortions at the edges of distributed actuators are neglected due to ratio of actuator thickness over structure depth is small (>10)
- The actuator material is stiffer than the structure material, but the additional stiffness is neglected in the controlled composite structure in order to yield conservative results (the accuracy of the results are suitable for future investigations)
- All supports are idealised which conforms strictly to assigned nodal freedom at the support nodes
Optimisation:
- Objective function features the sum of squared errors at the 5 d.o.f. of all nodes between the achieved, desired and normalised shapes. An empirical process ensures that the loads for each structure investigated falls into intuitively correct values (fitness values at higher strains indicate more need of smart control at higher voltages with the same actuators)
- The smart control project is a typical multivariate, static, non-linear programming problem with implicit function (fitness / errors / deviations / damping / simulated numerical iterative response)
- Discrete solutions are required in the form of constraining each actuator pair patch to be placed elementwise, with the shape & size being restricted to the shape & size of the elements bonded (each patch must occupy multiples of 1 element of the mesh adopted for FEM modelling)
- Voltage constraint is the breakdown voltage assumed to be 1000V/mm of piezoceramic thickness, thus with a 0.1mm thick patch, the voltage range is [-100V,100V]
- In multiple-patch control, overlapping of patches is not allowed as the structure is assumed to be controlled by only one actuator at any one location on the plan
- Genetic processes are performed on a structured random basis within the domain defined by all constraints
Genetic algorithms:
- The conceptual scheme for genetic algorithms as introduced by J.H. Holland (1) 1975 is modified, designed and implemented successfully
- Main processes of initial population, fitness evaluation, parent selection, crossover, mutation and replacement are assumed with customisation occurring within each process for project needs
- Multiples of 9 is applicable for number of chromosomes within each population to be tested due to the constraint of Regionalisation design selected for this project
- Termination of the optimisation occurs in 2 ways - either by termination at the maximum generation or by convergence parameters set by the end user (Vtolerance, number of repetition of the same best chromosomes in consecutive populations)
- The random generator forms the basis of simulated evolution within the genetic algorithms - for all processes and details, but structured for efficiency and geared to the needs of the project & end user
- 2 reproduction schemes - Regionalisation (splitting into distinct, ranked dynamic regions for structured optimisation) & Simplex (retention of best chromosome with all odd chromosomes being its mutations for exploitation & with all even chromosomes using original genetic processes for exploration) are tested & emerged as suitors
- Exploitation of the placement and voltage vicinity of the best chromosome of each population is added for improved efficiency
- Voltage exploration occurs in 2 distinct phases - using voltage step (Vstep) of 10 decreasing to 1 for pinpointing targets & voltage genetics for evolution of optimal voltage simultaneous with placement optimisation