Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

A Deep Learning Approach to Shape Optimization Problems for Flexoelectric Materials Using the Isogeometric Finite Element Method

A new approach for flexoelectricmaterial shape optimization is proposed in this study.In this work,a proxymodel based on artificial neural network(ANN)is used to solve the parameter optimization and shape optimization problems.To improve the fitting ability of the neural network,we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training.The isogeometric analysis-finite element method(IGA-FEM)is used to discretize the flexural theoretical formulas and obtain samples,which helps ANN to build a proxy model from the model shape to the target value.The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

A Subdivision-Based Combined Shape and Topology Optimization in Acoustics

We propose a combined shape and topology optimization approach in this research for 3D acoustics by using the isogeometric boundary element method with subdivision surfaces.The existing structural optimization methods mainly contain shape and topology schemes,with the former changing the surface geometric profile of the structure and the latter changing thematerial distribution topology or hole topology of the structure.In the present acoustic performance optimization,the coordinates of the control points in the subdivision surfaces fine mesh are selected as the shape design parameters of the structure,the artificial density of the sound absorbing material covered on the structure surface is set as the topology design parameter,and the combined topology and shape optimization approach is established through the sound field analysis of the subdivision surfaces boundary element method as a bridge.The topology and shape sensitivities of the approach are calculated using the adjoint variable method,which ensures the efficiency of the optimization.The geometric jaggedness and material distribution discontinuities that appear in the optimization process are overcome to a certain degree by the multiresolution method and solid isotropic material with penalization.Numerical examples are given to validate the effectiveness of the presented optimization approach.

A Non-parametric Gradient-Based Shape Optimization Approach for Solving Inverse Problems in Directed Self-Assemblyof Block Copolymers

We consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts.Specifically,we model polymer selfassembly using the self-consistent field theory and derive,in a non-parametric setting,the sensitivity of the dissimilarity between the desired and the actual morphologies to arbitrary perturbations in the guiding pattern shape.The sensitivity is then used for the optimization of the confining pattern shapes such that the dissimilarity between the desired and the actual morphologies is minimized.The efficiency and robustness of the proposed gradient-based algorithm are demonstrated in a number of examples related to templating vertical interconnect accesses(VIA).

Physical-Informed Neural Networks (PINNs) for Solving Shape Optimization Problems

In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.

INTEGRATION SHAPE AND SIZING OPTIMIZATION OF COMPOSITE WING STRUCTURE BASED ON RESPONSE SURFACE METHOD

An effective optimization method for the shape/sizing design of composite wing structures is presented with satisfying weight-cutting results. After decoupling, a kind of two-layer cycled optimization strategy suitable for these integrated shape/sizing optimization is obtained. The uniform design method is used to provide sample points, and approximation models for shape design variables. And the results of sizing optimization are construct- ed with the quadratic response surface method （QRSM）. The complex method based on QRSM is used to opti- mize the shape design variables and the criteria method is adopted to optimize the sizing design variables. Compared with the conventional method, the proposed algorithm is more effective and feasible for solving complex composite optimization problems and has good efficiency in weight cutting.

An Improved JSO and Its Application in Spreader Optimization of Large Span Corridor Bridge

In this paper,given the shortcomings of jellyfish search algorithmwith low search ability in the early stage and easy to fall into local optimal solution,this paper introduces adaptive weight function and elite strategy,improving the global search scope in the early stage and the ability to refine the local development in the later stage.In the numerical study,the benchmark problem of dimensional optimization with a 10-bar truss structure and simultaneous dimensional shape optimization with a 15-bar truss structure is adopted,and the corresponding penalty method is used for constraint treatment.The test results show that the improved jellyfish search algorithm can provide better truss sections as well as weights.Because when the steel main truss of the large-span covered bridge is lifted,the site is limited and the large lifting equipment cannot enter the site,and the original structure does not meet the problem of stress concentration and large deformation of the bolt group,so the spreader is used to lift,and the improved jellyfish search algorithm is introduced into the design optimization of the spreader.The results show that the improved jellyfish algorithm can efficiently and accurately find out the optimal shape and weight of the spreader,and throughMidas Civil simulation,the spreader used canmeet the requirements of weight and safety.

Study on Adaptive Shape Optimization

Aim To introduce a new method of adaptive shape optimization (ASOP) based on three-dimensional structure boundary strength and optimize an engine bearing cap with the method. Methods Using the normal substance's property of thermal expansion and cooling shrinkage,the load which is proportional to the difference between the nodes' stress and their respective objective stress were applied to the corresponding variable nodes on the boundary.The thermal load made the nodes whose stress is greater than their objective stress expand along the boundary's normal direction and the nodes whose stress is less than objec- tive stress shrink in the opposite direction , This process would repeat until the stress on the boundary nodes was converge to the objective stress. Results The satisfied results have been obtained when optimizing an engine bearing cap.The mass of the bearing cap is reduced to 55 percent of the total. Conclusion ASOP is an efficient,practical and reliable method which is suitable for optimizing the shape of the continuous structures.

Research on shape optimization of CSG dams

The multi-objective optimization method was used for shape optimization of cement sand and gravel （CSG） dams in this study. The economic efficiency, the sensitivities of maximum horizontal displacement and maximum settlement of the dam to water level changes, the overall stability, and the overall strength security were taken into account during the optimization process. Three weight coefficient selection schemes were adopted to conduct shape optimization of a dam, and the case studies lead to the conclusion that both the upstream-and downstream dam slope ratios for the optimal cross-section equal 1：0.7, which is consistent with the empirically observed range of 1：0.6 to 1;0.8 for the upstream and downstream dam slope ratios of CSG dams. Therefore, the present study is of certain reference value for designing CSG dams.

SIMULTANEOUS SHAPE AND TOPOLOGY OPTIMIZATION OF TRUSS UNDER LOCAL AND GLOBAL STABILITY CONSTRAINTS

A new approach for the solution of truss shape and topology optimization problems under local and global stability constraints is proposed.By employing the cross sectional areas of each bar and some shape parameters as topology design variables,the difficulty arising from the jumping of buckling length phenomenon can be easily overcome without the necessity of introduc- ing the overlapping bars into the initial ground structure.Therefore computational efforts can be saved for the solution of this kind of problem.By modifying the elements of the stiffness matrix using Sigmoid function,the continuity of the objective and constraint functions with respect to shape design parameters can be restored to some extent.Some numerical examples demonstrate the effectiveness of the proposed method.

A Double-Stage Surrogate-Based Shape Optimization Strategy for Blended-Wing-Body Underwater Gliders

In this paper,a Double-stage Surrogate-based Shape Optimization(DSSO)strategy for Blended-Wing-Body Underwater Gliders(BWBUGs)is proposed to reduce the computational cost.In this strategy,a double-stage surrogate model is developed to replace the high-dimensional objective in shape optimization.Specifically,several First-stage Surrogate Models(FSMs)are built for the sectional airfoils,and the second-stage surrogate model is constructed with respect to the outputs of FSMs.Besides,a Multi-start Space Reduction surrogate-based global optimization method is applied to search for the optimum.In order to validate the efficiency of the proposed method,DSSO is first compared with an ordinary One-stage Surrogate-based Optimization strategy by using the same optimization method.Then,the other three popular surrogate-based optimization methods and three heuristic algorithms are utilized to make comparisons.Results indicate that the lift-to-drag ratio of the BWBUG is improved by 9.35%with DSSO,which outperforms the comparison methods.Besides,DSSO reduces more than 50%of the time that other methods used when obtaining the same level of results.Furthermore,some considerations of the proposed strategy are further discussed and some characteristics of DSSO are identified.

Global Continuity Adjustment and Local Shape Optimization Technique for Complex Trimmed Surface Model

Smoothly stitching multiple surfaces mainly represented by B-spline or NURBS together is an extremely important issue in complex surfaces modeling and reverse engineering. In recent years, a lot of progress has been made in smooth join of non-trimmed surface patches, while there has been seldom research on smoothly stitching trimmed surface patches together. This paper studies the problem of global continuity adjustment, damaged hole repair and local shape optimization for complex trimmed surface model, and presents a uniform scheme to deal with continuity adjustment of trimmed surfaces and geometric repair of local broken region. Constrained B-spline surface refitting technique and trim calculation are first utilized to achieve global G^1 continuity, and then local shape optimization functional is adopted to reduce fitting error and improve local quality of refitted surface patch. The proposed approach is applied to a discontinuity ship hull surface model with an irregular hole, and the result demonstrates the validation of our method. Furthermore, on the premise of global continuity, the proposed locally repairing damaged surface model provides a better foundation for following research work, such as topology recovery technique for complex surface model after geometric repair.

Mechanical Behavior and Shape Optimization of Lining Structure for Subsea Tunnel Excavated in Weathered Slot

Subsea tunnel lining structures should be designed to sustain the loads transmitted from surrounding ground and groundwater during excavation. Extremely high pore-water pressure reduces the effective strength of the country rock that surrounds a tunnel, thereby lowering the arching effect and stratum stability of the structure. In this paper, the mechanical behavior and shape optimization of the lining structure for the Xiang＇an tunnel excavated in weathered slots are examined. Eight cross sections with different geometric parameters are adopted to study the mechanical behavior and shape optimization of the lining structure. The hyperstatic reaction method is used through finite element analysis software ANSYS. The mechanical behavior of the lining structure is evidently affected by the geometric parameters of crosssectional shape. The minimum safety factor of the lining structure elements is set to be the objective function. The efficient tunnel shape to maximize the minimum safety factor is identified. The minimum safety factor increases significantly after optimization. The optimized cross section significantly improves the mechanical characteristics of the lining structure and effectively reduces its deformation. Force analyses of optimization process and program are conducted parametrically so that the method can be applied to the optimization design of other similar structures. The results obtained from this study enhance our understanding of the mechanical behavior of the lining structure for subsea tunnels. These results are also beneficial to the optimal design of lining structures in general.

Reliability-Based Shape Optimization of Tenon/Mortise in Aero-engines with Contact

Based on the theory of reliability-based structural shape optimization, exact expressions of the sensibility using the stochastic finite element method for contact problems were derived in detail, and the basic steps of structural optimization were given. A coattail-type tenon/mortise of an aero-engine was optimized. In this model, the maximum equivalent stress of the nodes on the boundary of the tenon was the objective function; the width of tooth’s neck and the side surface’s slope angle of a tenon were design variables, with constraints of tension stress, extrusion stress and reliability index. The result showed that the distributions of the contact pressure between tenon and mortise, the equivalence stress and reliability index were more reasonable. It validates the correctness of the optimization model and the reliability-based structural shape optimization, and provides valuable references for structural design of the tenon/mortise.

Optimum Design of Structure Shape for Offshore Jacket Platforms

With the introduction of the design variables of nodal coordinates, which reflect the embedded depth of the pile and the jacket bed height, a shape optimum design model for offshore jacket platforms is established. A sequential two-level optimum algorithm is developed based on the design variable gradation. On the basis of the finite element method, the sensitivity of the objective function and nodal displacement is analyzed. As an example, the BZ281 oil storage offshore platform, which ties in the Bohai oil field, is designed with the shape optimum model. The results are compared with the cross-section optimum design. The tendency of design variables and its reasons in the two methods are analyzed. In the shape optimum design, the value of objective function is obviously smaller than that of the initial design and the cross-section optimum design. Therefore, the advantage of structure shape optimum design for jacket platforms is remarkable.

Permanent Magnet Shape Optimization Method for PMSM Air Gap Flux Density Harmonics Reduction

This paper proposed a permanent magnet optimization method to suppress the air gap flux density harmonic of permanent magnet synchronous motor(PMSM).The method corrected the effective air gap length of the motor,calculated the magnetization length of the permanent in the case of parallel magnetization,and took the influence of the permanent magnet relative permeability into consideration.Based on these works,for a given sinusoidal air gap flux density waveform,the corresponding structural parameters can be calculated,so as to achieve the optimization of the permanent magnet.By using this method to optimize the shape of the magnet,the fundamental wave of the air gap flux density can be retained to the greatest extent,so as to eliminate harmonics and maintain the output capacity at the same time.The feasibility and accuracy of the method have been verified by finite element analysis(FEA)and prototype machine experiment.This method is simple and time-saving,and has a satisfactory accuracy,which provides a reference method for permanent magnet optimization of PMSM.

A biarc-based shape optimization approach to reduce stress concentration effects

In order to avoid stress concentration, the shape boundary must be properly designed via shape optimiza- tion. Traditional shape optimization approach eliminates the stress concentration effect by using free-form curve to present the design boundaries without taking the machin- ability into consideration. In most numerical control （NC） machines, linear as well as circular interpolations are used to generate the tool path. Non-circular curves, such as non- uniform rotational B-spline （NURBS）, need other more ad- vanced interpolation functions to formulate the tool path. Forming the circular tool path by approximating the opti- mal free curve boundary with arcs or biarcs is another op- tion. However, these two approaches are both at a cost of sharp expansion of program code and long machining time consequently. Motivated by the success of recent researches on biarcs, a reliable shape optimization approach is pro- posed in this work to directly optimize the shape boundaries with biarcs while the efficiency and precision of traditional method are preserved. Finally, the approach is validated by several illustrative examples.

Convergence of shape optimization calculations of mechanical components using adaptive biological growth and iterative finite element methods

Shape optimization of mechanical components is one of the issues that have been considered in recent years. Different methods were presented such as adaptive biological for reducing costs and increasing accuracy. The effects of step factor, the number of control points and the definition way of control points coordinates in convergence rate were studied. A code was written using ANSYS Parametric Design Language (APDL) which receives the studied parameters as input and obtains the optimum shape for the components. The results show that for achieving successful optimization, step factor should be in a specific range. It is found that the use of any coordinate system in defining control points coordinates and selection of any direction for stimulus vector of algorithm will also result in optimum shape. Furthermore, by increasing the number of control points, some non-uniformities are created in the studied boundary. Achieving acceptable accuracy seems impossible due to the creation of saw form at the studied boundary which is called "saw position".

Construction of Design Guidelines for Optimal Automotive Frame Shape Based on Statistical Approach and Mechanical Analysis

A body frame composed of thin sheet metal is a crucial structure that determines the safety performance of a vehicle.Designing a correct weight and high-performance automotive body is an emerging engineering problem.To improve the performance of the automotive frame,we attempt to reconstruct its design criteria based on statistical and mechanical approaches.At first,a fundamental study on the frame strength is conducted and a cross-sectional shape optimization problem is developed for designing the cross-sectional shape of an automobile frame having a very high mass efficiency for strength.Shape optimization is carried out using the nonlinear finite element method and a meta-modeling-based genetic algorithm.Data analysis of the obtained set of optimal results is performed to identify the dominant design variables by employing the smoothing spline analysis of variance,the principal component analysis,and the self-organizing map technique.The relationship between the cross-sectional shape and the objective function is also analyzed by hierarchical clustering.A design guideline is obtained from these statistical approach results.A comparison between the statistically obtained design guideline and the conventional one based on the designers’experience is performed based on mechanical interpretation of the optimal cross-sectional frame.Finally,a mechanically reasonable new general-purpose design guideline is proposed for the cross-sectional shape of the automotive frame.

Shape Optimization for A Conventional Underwater Glider to Decrease Average Periodic Resistance

As a type of autonomous underwater vehicle(AUV),underwater gliders(UG)are getting increasing attention in ocean exploration.To save energy and satisfy the mission requirements of a longer voyage,shape optimization for UGs has become a key technique and research focus.In this paper,a conventional UG,including its fuselage and hydrofoil,is optimized,which aims to decrease the average resistance in one motion cycle.To operate the optimization progress for the complex object,multiple free form deformation(FFD)volumes are established for geometric parameterization.High-fidelity simulation models are employed for objective function evaluation and gradients calculation.And sequential quadratic programming(SQP)method is adopted as an optimization algorithm.The optimization results show that there exists a UG with symmetrical and non-horizontal hydrofoils that has lower resistance.