Large Number Of Design Variables Research Articles

Design of a low noise turbofan duct via an acoustic gradient-enhanced Kriging method

The Kriging models which are frequently used in aerodynamic shape optimization may become computationally inefficient when solving problems with large numbers of design variables. One solution to this problem would be the application of gradient-enhanced Kriging model. A gradient-enhanced Kriging and acoustic adjoint-method approach to duct acoustic problems is developed, aimed to improve the efficiency and accuracy of the existing Kriging approach at acoustic problems with many design parameters. To our knowledge, it is the first application of gradient-enhanced Kriging for duct acoustic problem. It employs a Kriging response surface in the parameter space, augmented with gradients obtained from the acoustic adjoint equations efficiently. The present paper aims at describing the potential of the gradient-enhanced Kriging method for low noise turbofan duct design. Prior to the optimization process, the implementation of the unsteady aeroacoustic adjoint method in shape optimization is validated by comparing the gradient values with that obtained by finite differences. In this work, the ordinary Kriging model and gradient-enhanced Kriging method are applied firstly to a benchmark functions and the results show that the additional gradient information can significantly enhance the accuracy of Kriging model. And then, the original Kriging-based, adjoint-based and the gradient-enhanced Kriging method are all used to model 50 variable duct acoustic problems, respectively. The test results show that this approach whose gradient information is introduced by using acoustic adjoint method developed from multimode LEE, named as acoustic gradient-enhanced Kriging, can significantly enhance the accuracy of Kriging models when the gradient data are available and thus provide an optimized low noise intake while maintaining the aerodynamic performance.

Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.

Multiobjective discrete optimization using the TOPSIS and entropy method for protection of pedestrian lower extremity

Design of vehicle frontal structures signifies an important topic of studies on protection of pedestrian lower extremity. Conventional optimization has been largely focused on continuous variable problems without involving complexity of human model. Nevertheless, design of frontal structures is commonly discrete from a manufacturing perspective and the responses of the lower extremity of pedestrian are highly nonlinear in nature. For this reason, this study aimed to develop a multiobjective discrete optimization (MODO) algorithm for design of frontal structures involving a pedestrian model. In the proposed MODO method, the order preference by similarity to ideal solution (TOPSIS) was coupled with the entropy algorithm to develop a multiple attribute decision making (MADM) model for converting multiple conflicting objectives into a unified single cost function. The presented optimization procedure is iterated using the successive orthogonal experiment to deal with a large number of design variables and design levels. In this study, the proposed method was first verified by two benchmark examples; and then was applied to a more sophisticated real-life design case of vehicle frontal structure to better protect pedestrian lower extremity from impacting injury. The finite element model was validated via experimental tests first and the optimized design was then prototyped for the further physical tests. The results showed that the algorithm is able to achieve an optimal design in a fairly efficient manner. The proposed algorithm exhibits considerable potential to solve other complex engineering design problems.

Distribution Optimization for Acoustic Design of Porous Layer by the Boundary Element Method

In this work, we develop an optimization approach to optimize the distribution of porous material layer inside cavity. The optimization seeks to improve the absorbing effects of the porous material, decreasing the noise level at regions of interest or increasing the sound energy dissipated by the porous material. To achieve the preset optimization aim, two different objective functions are accordingly defined. The acoustic absorption characteristics of porous materials are numerically described using the Delany–Bazley–Miki empirical model and modeled by the admittance boundary conditions in the boundary element analysis. Based on the solid isotropic material with penalization method, an admittance interpolation scheme is established between the element admittance and artificial element density. This transforms the discrete optimization into a continuous optimization problem, which can be solved by a gradient solver with the sensitivity information. As a key treatment in this study, we develop a fast sensitivity analysis approach based on an adjoint variable method and the fast multipole method to calculate the sensitivities of the objective function with respect to a large number of design variables. Finally, we validate the proposed optimization approach through a cabin example.

Structural Topology Design Optimization Using the Binary Bat Algorithm

In this study, the binary bat algorithm (BBA) for structural topology optimization is implemented. The problem is to find the stiffest structure using a certain amount of material and some constraints using the bit-array representation method. A new filtering algorithm is proposed to make BBA find designs with no separated objects, no checkerboard patterns, less unusable material, and higher structural performance. A volition penalty function for topology optimization is also proposed to accelerate the convergence toward the optimal design. The main effect of using the BBA lies in the fact that the BBA is able to handle a large number of design variables in comparison with other well-known metaheuristic algorithms. Based on the numerical results of four benchmark problems in structural topology optimization for minimum compliance, the following conclusions are made: (1) The BBA with the proposed filtering algorithm and penalty function are effective in solving large-scale numerical topology optimization problems (fine finite elements mesh). (2) The proposed algorithm produces solid-void designs without gray areas, which makes them practical solutions that are applicable in manufacturing.

DAFoam: An Open-Source Adjoint Framework for Multidisciplinary Design Optimization with OpenFOAM

The adjoint method is an efficient approach for computing derivatives that allow gradient-based optimization to handle systems parameterized with a large number of design variables. Despite this advantage, implementing the adjoint method for a partial-differential-equation-based primal solver is a time-consuming task. To lower the barrier for adjoint implementations, an object-oriented framework (DAFoam) is proposed to rapidly implement the discrete adjoint method for any steady-state OpenFOAM primal solver by adding or modifying only a few hundred lines of source code. In this paper, the DAFoam framework is introduced and the proposed object-oriented adjoint development process is illustrated. Using this strategy, the adjoint method is implemented for eight primal solvers, five turbulence models, and one radiation model in OpenFOAM. Excellent adjoint speed and scalability, with up to 10 million cells and 1536 CPU cores, and an average error in the adjoint derivatives of less than 0.1% are achieved. Finally, the implemented adjoint solvers and models are integrated into a gradient-based optimization framework, and four distinct design optimizations (multipoint aerodynamic optimization of a low-speed unmanned-aerial-vehicle wing, aerodynamic optimization of a transonic aircraft configuration, aerothermal optimization of a turbine internal cooling passage, and aerostructural optimization of a compressor rotor) are shown. DAFoam is available under an open-source license and is a powerful tool for the high-fidelity multidisciplinary design optimization of engineering systems, such as aircraft, ground vehicles, marine vessels, and turbomachinery.

Investigating the Reliability of Machine Learning Algorithms for Inverse Design: Polarization Rotator Unit-Cell Model

In this article, we implement three types of data-driven algorithms for the inverse design of the polarization rotator (PR) unit cell. Toward this end, a novel configuration pattern of the PR unit cell is proposed, which is reshapable to different geometries with a large number of design variables and recasts as a regression problem. The state-of-the-art algorithms including the neural network (NN), the deep neural networks (DNNs) with multiple hidden layers, and the support vector regression (SVR) are experimented with cross validations for ensuring the prediction generalizability. Averaged over all the experiments with competitive performances, the highest prediction accuracy about 95.23% was achieved for the SVR algorithm. This demonstrates the enormous capability of the data-driven algorithms in the geometrical dimension prediction of the unit cells for any given frequency band designated in the radar range (X, Ku, K, and Ka). The proposed inverse design procedure can expedite, facilitate, and to some extent replace the conventional and time-consuming manual design approaches with electromagnetic (EM) simulation software. Although these models can be very efficient in practice, they might be vulnerable against adversarial attacks that craft fake inputs to purposely fool the victim regressors toward adversary's wishes. This poses security concerns for the learning-based algorithms and might negatively affect their prediction reliabilities in runtime. In this article, we only characterize the existence of the adversarial attacks for the regression models using the fast gradient sign method. Our conducted experiments uncover that the fooling rate of all the aforementioned cutting-edge NN and DNN-based regressors is above 98% and this rate for the SVR model is about 11% better than other models.

Massive simplification of the wind farm layout optimization problem

Abstract. The wind farm layout optimization problem is notoriously difficult to solve because of the large number of design variables and extreme multimodality of the design space. Because of the multimodality of the space and the often discontinuous models used in wind farm modeling, the wind industry is heavily dependent on gradient-free techniques for wind farm layout optimization. Unfortunately, the computational expense required with these methods scales poorly with increasing numbers of variables. Thus, many companies and researchers have been limited in the size of wind farms they can optimize. To solve these issues, we present the boundary-grid parameterization. This parameterization uses only five variables to define the layout of a wind farm with any number of turbines. For a 100-turbine wind farm, we show that optimizing the five variables of the boundary-grid method produces wind farms that perform just as well as farms where the location of each turbine is optimized individually, which requires 200 design variables. Our presented method facilitates the study and both gradient-free and gradient-based optimization of large wind farms, something that has traditionally been less scalable with increasing numbers of design variables.

Crashworthiness optimization of VRB thin-walled structures under manufacturing constraints by the eHCA-VRB algorithm

Variable-thickness rolled blanks (VRBs) represent an important approach for constructing lightweight structures. However, the optimization of the crashworthiness and thickness distribution of VRB thin-walled structures under manufacturing constraints is a nonlinear dynamic-response structural-optimization problem that has a large number of design variables. To tackle this problem, this paper has extended and improved the hybrid cellular automaton for thin-walled structures (HCATWS) algorithm, and has proposed an extended hybrid cellular automaton for VRB thin-walled structures (eHCA-VRB) algorithm. This algorithm consists of an outer loop and an inner loop. The outer loop performs crash simulation analysis to define an appropriate target mass for the inner loop, whereas the inner loop adjusts cell thicknesses according to the internal energy density (IED) of the current cell and its neighboring cells so that the IED in the design domain becomes evenly distributed. A one-dimensional CA model is defined along with the rolling direction based on the thickness distribution of VRB thin-walled structures. Furthermore, the eHCA-VRB algorithm also generates a mapping relationship between the one-dimensional CA model and the FE model. To optimize the thickness distribution of VRB thin-walled structures under manufacturing constraints, our method uses cell thickness as a design variable and incorporates the constraints of the VRB rolling process in the cell thickness update rules. To verify the convergence and efficiency of the eHCA-VRB algorithm, VRB top-hat thin-walled structures are optimized for crashworthiness with/or without manufacturing constraints (M.C.), respectively. The results show that the eHCA-VRB algorithm can be used to efficiently solve the optimization problems of crashworthiness and the thickness distribution of VRB thin-walled structures under manufacturing constraints.

Optimization of building form to reduce incident solar radiation

In recent years, concerns about global warming have encouraged researchers to incorporate optimization techniques into the design of energy-efficient buildings. Designing a building with energy consumption in mind, design teams should apply the appropriate considerations at the early architectural conceptual design stage. Furthermore, constructing buildings that are energy-efficient results in using less natural resources to cool buildings. Since the impact of solar radiation is a key factor in building energy, reducing insolation (incident solar radiation) is critical for energy-efficient buildings. To alleviate the energy concerns in energy-efficient buildings, this paper proposes an optimization approach to study energy-efficient building form that minimizes insolation while preserving the required total built area. Furthermore, a software system is built with the optimization in the back end and a user interface for experimenting with different design parameters and visualizing the resulting building form. The tool can be used by architectural design teams to design energy-efficient buildings. Due to the large number of design variables in a building and the nonlinear constrained characteristics, the optimization uses the computationally efficient Penalty Successive Linear Programming (PSLP) technique. PSLP is used to solve the large nonlinear optimization problem via a sequence of Linear Programs (LPs) to generate the optimized building form with low computational expense. The optimization and software system are illustrated using real scenarios from several countries while considering different building laws. A complete validation process is undertaken using the CPLEX and MATLAB software. Results show that the optimization tool provides up to 48% less insolation while still meeting different site and building constraints.

Aerodynamic/aeroacoustic variable-fidelity optimization of helicopter rotor based on hierarchical Kriging model

Rotor noise is one of the most important reasons for restricting helicopter development; hence, the optimization design of rotor blade considering aeroacoustic and aerodynamic performance at the same time has always been the focus of research attention. For complex rotor design problems with a large number of design variables, the efficiency of the traditional Kriging model needs to be improved. Thus, Hierarchical Kriging (HK) model is employed in this study for rotor optimization design. By using the validated RANS solver and acoustic method based on the FW–Hpds equation, an efficient aerodynamic/aeroacoustic optimization method for high-dimensional problem of rotors in hover based on HK model is developed. By using present HK model and new infill-sampling criteria, the number of design variables is increased from less than 20–53. Results of two analytical function test cases show that the HK model is efficient and accurate in calculation. Subsequently, the helicopter rotor blade is optimally designed for aerodynamic/aeroacoustic performance in hover based on the HK model with high dimensional design variables. The objective function is adopted to improve the rotational noise characteristics by reducing the absolute peak of the acoustic pressure. In addition, the constraints of thrust, hover efficiency, solidity, and airfoils thickness are strictly satisfied. Optimization results show that the Kriging model finds the objective of reducing the noise by 2.87 dB after 248 iterations while the HK model does it only after 164 iterations. The optimization efficiency of the HK model is significantly higher than that of the traditional Kriging model. In the case analyzed, the HK model saves 35% of the time used by the Kriging model.

Coordinated optimal design of zero/low energy buildings and their energy systems based on multi-stage design optimization

As promising means to reduce carbon emission and energy consumption, zero/low energy buildings have been attracting increasing attentions. Multi-stage design optimization methods have been developed for zero/low energy buildings and their energy systems especially when a large number of design variables need to be optimized. However, these methods ignore the interactions between building envelope and energy system design optimizations, which can be addressed by simultaneous optimization methods requiring huge computation cost. In this study, a coordinated optimal design method is proposed as a computation cost-effective method for stand-alone and grid-connected zero/low energy buildings and their energy systems on the basis of multi-stage design optimization methods, to effectively consider the interactions between building envelope and energy system design optimizations. An iterative approach is adopted to coordinate multi-stage optimizations of building envelope and energy systems. The Hong Kong zero carbon building is used as the reference building to test and validate the proposed method. The results and experiences of the case studies show that the proposed coordinated design method can provide global optimal designs efficiently and robustly. The life cycle “cost” of the optimal designs is 4% less and unmet cooling load is over 22% less compared with existing multi-stage design methods.

Design optimization for self-propulsion of a bulk carrier hull using a discrete adjoint method

Computational fluid dynamics (CFD) based optimization is becoming increasingly popular in hydrodynamic design of ship hulls because it provides a fully automatic framework with a shorter design cycle than a human-supervised design tool. Despite the above advantage, CFD-based optimization requires careful attention to relevant design considerations, such that the final design is useful in practice. These considerations include all relevant objectives (such as drag and wake distortion) and constraints (such as volume, thickness, and curvature). Although constraints have been included in previous hull shape optimization studies, these studies have typically considered only one objective. To address this shortcoming, we conduct design optimization for self-propulsion by simultaneously considering drag and propeller-wake distortion. We use a gradient-based optimization framework that includes a discrete adjoint method for efficient derivative computation, which allows us to use a large number of design variables to parameterize the complex hull shape and thus gain a large amount of freedom for geometric modification. We impose appropriate geometric constraints (volume, thickness, and curvature) on the hull surface to ensure a practical design. In addition, we use a weighted objective function that includes drag and wake distortion to construct a Pareto front with five optimizations. We also consider hull-propeller interaction by comparing optimization results with and without a propeller. We use the Japan bulk carrier (JBC) as the baseline model and focus on optimizing the stern region. We find that optimizing for only one objective results in a large penalty on the other objective, whereas a weighted objective balances the penalty and achieves simultaneous improvement in drag and wake distortion. Moreover, we observe that the suction effect of the propeller suppresses the flow separation near the bilge tube and smooths out the velocity distortion at the propeller plane; these are effects that would end up affecting the optimized shapes. Our results demonstrate that it is necessary to simultaneously consider drag and wake distortion in hull-shape-optimization studies, and that constrained shape optimization with a large number of design variables is possible with the discrete-adjoint method.

Rapid Prototyping of Variable Angle-Tow Composites

The recent development of new manufacturing techniques of composite structures, e.g., additive manufacturing (AM) techniques, allows for going beyond the classical design rules, thus leading the designer to find innovative and more efficient solutions like the variable angle-tow (VAT) composites. VAT composites allow taking advantage from the benefits related to the curvilinear fibre path in the most effective way, though their utilisation unavoidably implies an increased complexity of the design process. In fact, VAT composites are characterised by a large number of design variables involved at different scales. Accordingly, a dedicated multi-scale optimisation approach able to integrate both mechanical and technological requirements (to ensure the manufacturability of the solution) must be developed. In this work, the multi-scale two-level (MS2L) optimisation strategy for VAT laminates is used, for the first time, to design a VAT laminate by taking into account the manufacturing requirements related to the fused filament fabrication (FFF) and continuous filament fabrication (CFF) processes. To show the effectiveness of the MS2L strategy in terms of manufacturability of the optimised solutions, a prototype of the VAT plate is realised by developing an ad hoc CAD model interfaced with the CFF machine. The realised prototype matches very well the optimum fibre path resulting from the MS2L strategy and the importance of including technological constraints within the design process is highlighted.

Design optimization of mooring system: An application to a vessel-shaped offshore fish farm

Design optimization of mooring systems of offshore floating structures is a challenging task, partly because of the large number of design variables, complicated design constraints, nonlinear system behavior, and time-consuming numerical simulations. For engineering designs, efficient yet accurate approaches are needed. This paper proposes an integrated optimization methodology for design of mooring systems. The methodology integrates the design of experiments, screening analysis, time-domain simulations, and a metamodel-based optimization procedure. To demonstrate the methodology, the mooring system of a vessel-shaped offshore fish farm was designed considering the ultimate limit state. The fully-coupled numerical model includes a floater, flexible fish cages and a single-point mooring system. The Kriging metamodels were applied as surrogates for the responses of time-domain simulations. The optimal solutions were found by exploring the design space using a gradient-based search algorithm. Validations were performed on the metamodels and the global optimal solutions. The proposed methodology is also applicable to design optimization of other marine structures.

Single/two-objective aerodynamic shape optimization by a Stackelberg/adjoint method

ABSTRACTIn this article, the Stackelberg game strategy is coupled with the adjoint method and applied in single-objective and two-objective aerodynamic shape optimizations. In the proposed method, two types of player (leader and follower) are involved, and each of these players is responsible for the optimization of one objective function by adjusting a subset of design variables. A Stackelberg equilibrium is reached when the leader cannot improve his/her objective function further. Note that the success of the proposed method is highly dependent on the choice of a few influential factors, including the maximal number of iterations for each player, the splitting and mapping schemes of design variables, and the allocation strategies of objective functions to different players. Therefore, the impacts of these factors are firstly assessed by a two-objective optimization case of the NACA0012 airfoil, and some useful inferences are produced for the choice of these factors. After that, single-objective and two-objective aerodynamic optimizations of the RAE2822 airfoil and the ONERA M6 wing are conducted to verify the usefulness of these inferences, and to validate the efficiency and effectiveness of the proposed method in optimization problems with a complex numerical model or a large number of design variables.

A dual boundary element based implicit differentiation method for determining stress intensity factor sensitivities for plate bending problems

A novel methodology for determining Stress Intensity Factor (SIF) sensitivities for plate bending problems using the Dual Boundary Element Method (DBEM) is presented. The direct derivatives of the DBEM integral equations for plate bending have been derived for the first time and are used as part of a DBEM-based Implicit Differentiation Method (IDM or DBEM-IDM) for calculating the sensitivities of SIFs to changes in different geometric parameters such as crack length and crack rotation angle. The SIFs and their sensitivities are calculated using the J-integral and the derivative of the J-integral respectively. A numerical example featuring a thick plate subjected to membrane, bending, and pressure loads is presented. In the first half of the numerical example, the SIF sensitivities from the IDM are compared with those obtained from the more common, but relatively crude, Finite Difference Method (FDM or DBEM-FDM). Results show that the IDM is a significantly more efficient and robust alternative to the FDM. The accuracy of the FDM showed significant dependence on the step size used, necessitating a time-consuming optimisation procedure to determine the optimal step size. Once this optimal step size was found, both methods provided very similar results. As part of the second half of the numerical example, a demonstration of one possible application of the SIF sensitivities from the IDM is presented. This involved carrying out reliability analyses using the First-Order Reliability Method (FORM) with a large number of design variables.

Inverse parameter identification of n-segmented multilinear cohesive laws using parametric finite element modeling

Delamination in laminated composites are efficiently modelled with the cohesive zone model (CZM). The shape of the cohesive law becomes important when simulating delamination in material systems experiencing large scale fiber bridging, as several competing damage mechanisms occurs in the fracture process zone at multiple length scales. For this purpose a multilinear cohesive law has recently been developed in [1], which readily can be adapted to a variety of shapes. However, a key challenge in applying such cohesive laws is their model calibration, i.e. identification of parameters that define the shape of the cohesive law. In this work, a new methodology for experimental characterization of multilinear cohesive laws is proposed. The methodology is an inverse approach, which identifies cohesive laws by iteratively varying cohesive zone parameters using a gradient-based optimization scheme to minimize the error in structural response between a fracture mechanical experiment and a parametric finite element model. The method is demonstrated on a moment loaded double cantilever beam (DCB) specimen made of unidirectional glass-epoxy showing large-scale fiber bridging. Multilinear cohesive laws are characterized which result in an excellent agreement between the finite element simulation and the experiment. The results and sensitivity studies demonstrate the accuracy and robustness of the proposed methodology, even for a large number of design variables in the optimization problem.

Aerothermal optimization of a ribbed U-bend cooling channel using the adjoint method

Aerothermal optimization is a powerful technique for automating the design of turbine internal cooling passages, where both pressure loss and heat transfer are considered. Existing optimization studies commonly adopt gradient-free algorithms, which can handle only a few design variables. However, to enhance heat transfer, internal cooling designs use complex geometries consisting of ribbed serpentine channels, which need to be parameterized by using a large number of design variables. To address this need, we develop herein an approach for aerothermal optimization that uses a gradient-based optimizer in conjunction with a discrete adjoint method to efficiently compute the required gradients with respect to numerous design variables. We apply this approach to the design of a ribbed U-bend channel, which is representative of a section of turbine internal cooling passages. The objective function combines heat transfer and pressure loss as a weighted sum. We find the Pareto front for these two objectives by running five optimizations with different weights. We consider both a rib-free and a ribbed U-bend configuration. For the rib-free configuration, we use 113 design variables to parameterize the U-bend shape. We compare our optimization results with those from gradient-free methods and demonstrate that the proposed method leads to lower pressure loss while enhancing heat transfer. For the ribbed configuration, we use 146 design variables and allow the ribs to change their arrangement independently (shape, height, pitch, and angle). Each rib adopts a requisite arrangement to balance heat transfer and aerodynamics, depending on the local flow conditions. Optimizing the U-bend shape is shown to be more effective for improving overall heat transfer than optimizing the rib arrangement. However, optimizing ribs is more effective for improving local heat transfer. The results demonstrate that the proposed optimization framework has the potential to handle general turbine heat transfer designs, not only for internal cooling but also for other design problems, such as film cooling and jet impingement cooling.

A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients

Motivated by the applications of the node-based shape optimization problems, where various response evaluations are often considered in constrained optimization, we propose a modified search direction method for optimization design updates. There exist numerous methods for solving constrained optimization problems. The methods of the class active-set strategy try to travel along the active constraints to solve constrained optimization problems. Contrary to the active-set methods, the algorithms of the class interior-point method reach an optimal solution by traversing the interior of the feasible domain. This characteristic is considered to be beneficial for shape optimization because the usual zig-zagging behavior when traveling along the active constraints is avoided. However, the interior-point methods require the solution of a Newton problem in each iteration step, which is generally considered to be difficult for the shape optimization problem. In the present work, we propose a modified search direction method that applies the singular-value decomposition method for the normalized objective and constraint gradients. The modified search direction is a descent direction of the objective function. Using this search direction for the design update in each iteration, we observe that the centrality conditions for the interior-point method are approached iteratively. We are able to achieve a local minimum by traversing the interior of the feasible domain by only using the gradient information of the objective and constraint functions. The results are shown first with analytical 2D problems, and then the results of shape optimization problems with a large number of design variables are discussed. To robustly deal with complex geometries, the Vertex Morphing method is used.