Mathematical shape optimization of hip prosthesis design

IntroductionIn recent years, SMPMSM is quietly famous compared with other conventional motors in industrial applications due to higher efficiency, higher reliability, low cost, and smooth torque [1]–[3]. In general, several objective optimizations need to be consider, which required an optimization technique for an electrical motor. This paper deals with an optimal design of a surface-mounted permanent magnet synchronous machine (SMPMSM) with an exact analytical subdomain model by using a genetic algorithm method. To analyze the characteristic of permanent magnet (PM) motors, the classical optimization method, such as the finite element method (FEM), is intensively used. However, FEM has several time problems that require a longer computational time to evaluate the performance of PM motors. This problem can be overcome by using a genetic algorithm (GA) method combined with a subdomain model (SD), which developed an improved performance of SMPMSM, for instance, total harmonic distortion (THDv) and phase back-EMF. In this design, a three-phase 12-slot/8-pole PM motor is established with an exact SD model with radial magnetization patterns. Then, the GA ensemble with SD model to search the optimality of SMPMSM machine design. In the final analysis, the optimal new design of SMPMSM demonstrated by comparing with the initial design that is investigated by FEM. The result of induced back-EMF, and magnetic flux density of optimal design is compared with the initial design to show the advantages of GA optimization method.Analytical Model, Genetic Algorithm and Parameter of optimizationThe purpose of this analysis is to investigate the optimization design of a three-phase 12-slot/8-pole SMPMSM with a radial magnetization pattern. This optimization problem has been solved by obtains the minimum THDv as a fitness function. In order to process the optimization design of SMPMSM, several parameters need to be optimized i.e magnet pole-pitch, αp and slot-opening angle, boa. The roulette method is used to select the objective function of optimization with 50 maximum generations per iteration. The cross and mutation probability need to be set which requires linear and uniform recombination which is 0.89 and 0.025 respectively. The parameters of 12-slot/8-pole SMPM motor as follows: radius of the rotor, Rr = 23mm, the radius of PM surface Rm = 26mm, the inner radius of the stator Rsb = 42.5mm, axial length, la = 50mm, air-gap length, lg = 1mm, remanence of the magnet, Br, 1.12T, relative coil permeability, µr = 1.05 with the number of pole pairs, Np is 8 and number of stator slots, Ns is 12.The result, Discussion and ConclusionThe phase back electromotive force has a significant relationship with flux-linkage because phase back electromotive force is derived from the rate of change of flux-linkage. In Fig. 1. show the waveform of phase back electromotive force against rotor position of initial and optimal design in radial magnetization patterns. The initial design of phase back electromotive force waveform shows more trapezoidal compare with the optimal design which has a sinusoidal waveform. The comparison of total harmonic distortion between the initial and optimal design of SMPMSM are demonstrated in Fig. 2 below. The result from Fig.2. shows the highest total harmonic distortion of 12-slot/8-pole motor is an initial design which is 11.34% compared with the optimal design where 1.3%. The magnitude of harmonic in phase back electromotive force |En| of initial design shows 22.24 V compare to optimal design which is 17.4 V. This shows a good agreement that the optimal motor design gives lowest total harmonic distortion compare with the initial design. The optimizes of SMPMSM motor with the following parameters are given in Table 1. **

This paper is intended to overview on analytical and numerical methods in shape optimization. We compute and analyse the shape Hessian in order to distinguish well‐posed and ill‐posed shape optimization problems. We introduce different discretization techniques of the shape and present existence and convergence results of approximate solutions in case of well posedness. Finally, we survey on the efficient numerical solution of the state equation, including finite and boundary element methods as well as fictitious domain methods. Copyright © 2008 John Wiley & Sons, Ltd.

Structural topology optimization.- On basic properties of Michell's structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

Finite-element approximation of 2D elliptic optimal design

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

This research gauges the capabilities of deep reinforcement learning (DRL) techniques for direct optimal shape design in computational fluid dynamics (CFD) systems. It uses policy based optimization, a single-step DRL algorithm intended for situations where the optimal policy to be learnt by a neural network does not depend on state. The numerical reward fed to the neural network is computed with an in-house stabilized finite elements environment combining variational multi-scale modeling of the governing equations, immerse volume method, and multi-component anisotropic mesh adaptation. Several cases are tackled in two and three dimensions, for which shapes with fixed camber line, angle of attack, and cross-sectional area are generated by varying a chord length and a symmetric thickness distribution (and possibly extruding in the off-body direction). At a zero incidence, the proposed DRL-CFD framework successfully reduces the drag of the equivalent cylinder (i.e., the cylinder of same cross-sectional area) by 48% at a Reynolds numbers in the range of a few hundreds. At an incidence of 30°, it increases the lift to drag ratio of the equivalent ellipse by 13% in two dimensions and 5% in three dimensions at a chord Reynolds numbers in the range of a few thousands. Although the low number of degrees of freedom inevitably constrains the range of attainable shapes, the optimal is systematically found to perform just as well as a conventional airfoil, despite DRL starting from the ground up and having no a priori knowledge of aerodynamic concepts. Such results showcase the potential of the method for black-box shape optimization of practically meaningful CFD systems. Since the resolution process is agnostic to details of the underlying fluid dynamics, they also pave the way for a general evolution of reference shape optimization strategies for fluid mechanics and any other domain where a relevant reward function can be defined.

This paper presents design data defining optimal freeform notch shapes for use in tensile loading conditions for a range of geometries, notably the notch aspect ratio and plate widths. Here, an iterative two-dimensional finite element gradientless shape optimization procedure was used to obtain the optimal shapes, which offer the lowest possible stress concentration, subject to the geometric constraints. Precise non-dimensional coordinates are given, for a range of notch aspect ratios and plate widths, and are presented in a tabular form that allows them to be easily used by designers. All cases have been solved with a minimum radius of curvature constraint, which results in a ‘robust’ optimal shape that has no sharp corners. The optimal shapes were found to provide up to a 26 per cent reduction in local peak stress over circular shapes of the same aspect ratio. Optimal geometries are also given for limiting notches that provide a net section stress equal to unity, at a finite notch aspect ratio. The results obtained are applicable to both the initial design of notched components and the shape reworking of existing components to remove damage.

The study of reactive silencers by shape and parametric optimization techniques

Radar cross section (RCS) reduction technologies are very important in survivability of the military naval vessels. Ship appearance shaping as an effective countermeasure of RCS reduction redirects the scattered energy from one angular region of interest in space to another region of little interest. To decrease the scattering electromagnetic signals from ship scientifically, optimization methods should be introduced in shaping design. Based on the assumption of the characteristic section design method, mathematical formulations for optimal shaping design were established. Because of the computation-intensive analysis and singularity in shaping optimization, the response surface method (RSM) combined genetic algorithm (GA) was proposed. The polynomial response surface method was adopted in model approximation. Then genetic algorithms were employed to solve the surrogate optimization problem. By comparison RCS of the conventional and the optimal design, the superiority and effectiveness of proposed design methodology were verified.

Multi-objective free-form optimization for shape and thickness of shell structures with composite materials

The femoral component of the artificial hip joint implanted in a patient's femur is subjected to a complex set of forces exerted due to normal life activities. It is thought that high values of stress in the cement in a cemented prosthesis can lead to fractures of the cement mantle and loosening of the stem. The incidence of such problems may be diminished by reduction of the fatigue notch factor in the cement, such that stress concentrations are avoided and the crack initiation time maximised. This study describes a method for numerical shape optimisation to determine an optimal shape for the femoral stem of a hip prosthesis in order to minimise the fatigue notch factor in the cement layer at the interfaces with the bone and stem whilst at the same time maintaining or increasing the stress levels in the proximal medial femoral bone to help prevent stress shielding. The method when used to optimise the shape of a stainless steel Charnley stem was found to be extremely efficient and effective. The resulting optimal shape was heavily waisted in the proximal region below the neck but distally was quite similar to the original design. The fatigue notch factors in the cement were reduced by 16% and 19% for medial and lateral cement/stem interfaces, respectively, and by 8% and 2% for the corresponding cement/bone interfaces. The fatigue notch factor in the proximal medial bone was increased by 57% which indicates that the general stress level in this region is markedly increased. Thus the optimised design should increase the fatigue life of the cement and at the same time reduce stress shielding in the proximal bone. Both of these effects may help prevent loosening of the femoral component and hence reduce the need for early revision operations.

In today's society, with an increasing need for energy worldwide and at the same time a growing awareness of its negative consequences, optimizing and reducing energy consumption has become important for sustainability. The process of optimizing the appliances in large industry and household scale contributes to the minimization of the global energy consumption. In order to increase the efficiency of these appliances, various optimizations are employed, such as the shape optimization. In this thesis, a shape optimization method is used to modify the geometry of an existing heat exchanger. The heat exchanger is used in a domestic boiler and the shape optimization is a crucial step for its fast development. Currently, the shape of the heat exchanger is highly dependent on its manufacturing process. Thus, more complex shapes are investigated and their effects on the heat transfer and pressure drop are evaluated in this thesis. Two aspects need to be investigated for this heat exchanger optimization: the complex internal flow within the heat exchanger, and the performance of the optimization method as well as its resulting shapes. Since the heat exchanger consists of different flow regimes due to its complex geometry, a detailed study on these regimes (ReD = 2000, 100, and 10) is conducted prior to the optimization procedure. Furthermore, the complex geometry of this heat exchanger leads to different models used in this thesis. Therefore this thesis is divided into three parts based on the models: the three-dimensional single cylinder, the two-dimensional cylinder arrays, and the three-dimensional cylinder array. In optimizing the heat exchanger, the shape optimization method is justified as the most suitable method. Using the ANSYS Fluent's adjoint shape optimization method, the pressure drop and the heat transfer of the heat exchanger are optimized. This method has been used in the past for optimizing various geometries, yet to the author's knowledge, no detailed study on the cylinder array has been published. Consequently, a study on the adjoint parameters and its effect on the overall performance of the optimization procedure is of importance. Furthermore, best practices and recommendations for future work regarding the adjoint optimization procedure are elaborated.

An optimal shape of two-dimensional asymmetric diffuser with maximum pressure recovery at the exit is numerically obtained using a mathematical theory based on the variational calculus and gradient algorithm. The initial diffuser is taken to be a two-dimensional asymmetric diffuser for which much experimental and numerical data are available. The Reynolds number based on the bulk mean velocity and the channel height at the diffuser entrance is 1.8 × × 10 4 .F rom this initial shape, optimal diffuser shapes are designed for six different geometric constraints, such as the streamwise length and height of the diffuser. The optimality condition for maximum pressure recovery is obtained to be zero skin friction along the diffuser wall. The turbulent flow inside the diffuser is predicted using the k‐� ‐v2‐f model, and optimal shapes are obtained through iterative procedures to satisfy the optimality condition. With the shape design, flow separation that appeared in the initial diffuser is completely removed or significantly reduced. For one of the optimal diffuser shapes obtained, large-eddy simulation is carried out to validate the result of the shape design. The wall shear stress, wall pressure, mean velocity, and turbulence quantities obtained from large-eddy simulation are in good agreement with those from the simulation using the turbulence model.

Nowadays, weight reduction is great opportunity for automotive components design due to meet the structural strength without reducing the components safety. A classical try-and-error approach to design the automotive components in industries is inadequate which mean new methods are needed to enhance the design process. An improvement of the component can be achieved by adopting suitable optimization techniques at the early design stage. In this paper, the problem of automotive component design; steel wheel rim design in a view of weight reduction was tackled by means of shape optimization method. Design methodology was proposed to the original design of selected wheel rim and was continued optimized them by various reduction targets. The optimized design together with the datum has been analyzed and the results were compared and discussed. Shape optimization with 15% of reduction target at steel wheel rim was the best optimal design and it was met the design criteria and safety factor. By applying shape optimization, weight of steel wheel also has been reduced by using proper reduction target. The methodology has been proven to be successful in finding innovative and efficient layouts for automotive components design.

A shape optimization procedure is developed, using the boundary element method, for two-dimensional anisotropic structures to minimize weight while satisfying certain constraints upon stresses and geometry. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities of anisotropic materials. The boundary element method is very suitable for shape optimization and in comparison with the finite element method needs much fewer data, related only to the boundary of the structure being considered. Because a directly differentiated form of the boundary integral equation can be used to determine the derivatives of the objective and constraint functions, the accuracy of computation is very high. Because of the non-linear nature of weight and stresses, the numerical optimization method used in the program is the feasible direction approach, together with the golden section method for the one-dimensional search. Three example problems with anisotropic material properties are presented to demonstrate the applications of this general-purpose program.