Search

A unified shape and non-shape design optimization for lattice structures is performed to obtain extremal band gaps in an isogeometric computational framework, which uses the same NURBS basis functions as used in CAD description. Using a precise numerical model in both response and shape sensitivity analyses, enhanced response and sensitivity can be obtained, which leads to a precise optimal design. A plane wave propagation in infinite periodic lattice is analyzed within a representative unit cell using the Bloch theory with periodic boundary conditions. The ligaments of lattice structures are modeled using geometrically exact beams whose cross-sectional area and configuration are regarded as design variables, which are parameterized by the NURBS basis functions. The design variables are controlled by a mathematical programming algorithm with efficient adjoint design sensitivity. The obtained optimal lattice structures of triangular, squared, hexagonal, and Kagomé structures are fabricated using a 3D printing machine to be validated by physical experiments. Through harmonic response analysis and physical experiment for the optimal and the original designs, the wave attenuation properties are compared to demonstrate the validity of the obtained optimal design. [ABSTRACT FROM AUTHOR]

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method. • Derive continuum-based adjoint configuration design sensitivity for finite deformation beams. • Parameterization of finite rotation by an exponential map of skew-symmetric matrix. • Explicit parameterization of initial orthonormal frame by the smallest rotation method. • Material derivative of orthogonal transformation matrix at deformed configuration. [ABSTRACT FROM AUTHOR]

Isogeometric formulations of curved Timoshenko beams in curvilinear coordinates often result in self-straining of membrane and shear strains, due to the discretization of the local displacement field. Self-straining means a failure in exact representation of rigid body motions, which consequently deteriorates response quality. To overcome the difficulty of self-straining, we propose an invariant formulation that discretizes the global displacement field. It turns out that the approximated membrane, shear, and bending strain measures are invariant regardless of initial geometry in the proposed formulation. For effective applications to any arbitrarily curved structures and locking-free formulations to alleviate membrane and shear locking, the proposed invariant formulation is combined with selective reduced integration (SRI) and B ̄ projection method. Numerical examples demonstrate the effectiveness and applicability of the proposed invariant formulation, which gives much more accurate results together with both SRI and B ̄ projection method, in comparison to the existing isogeometric formulations. [ABSTRACT FROM AUTHOR]

We present an isogeometric shape optimization method for the design of magnetic actuators. The objective is to maximize the magnetic force of actuators, which tends to be highly dependent on the geometry around air gap. The isogeometric shape optimization is known to be robust for the actuator design, since it can preserve the exact geometry during the design optimization. Using the Maxwell stress tensor method, the magnetic force is calculated through the isogeometric analysis. The analytical design sensitivity provides geometric exactness due to the use of non-uniform rational B-splines basis functions. The optimization problem is solved using a gradient-based algorithm of modified method of feasible directions. To demonstrate the robustness of the isogeometric method, the geometric dependence of magnetic forces is investigated. In addition, the isogeometric optimization method is applied to the design problems of linear and rotating actuators, which yields the optimal shape of the ferromagnetic materials around the air gap. [ABSTRACT FROM AUTHOR]

This paper presents a variational method to update internal control points in isogeometric shape optimization. The important properties of domain parameterization such as bijective mapping between parametric and physical domains, uniform mesh, and orthogonal mesh are enforced simultaneously. The bijective mapping is achieved by minimizing a Dirichlet energy functional. To prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh, a constraint is introduced to enforce the positive Jacobian of mapping from parametric to physical domains. In spite of adding the constraint that might increase computational costs, the proposed method is more efficient due to the convexity of Dirichlet energy functional, compared with the other unconstrained methods. Also, it turns out that the proposed method is more effective to achieve the bijective mapping, especially near a concave boundary. The uniform parameterization of the domain is achieved by minimizing the Dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional. The required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points. The developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process. [ABSTRACT FROM AUTHOR]

An isogeometric analysis method is developed for the stress intensity factors in curved crack problems. In the isogeometric approach, NURBS (Non-Uniform Rational B-Spline) basis functions in CAD system is directly utilized in the response analysis, which enables the seamless incorporation of the higher continuity and the exact geometry such as curvature and normal vector into the computational framework. In mixed-mode curved crack problems, the precise evaluation of crack-face integral is essential to compute the precise stress intensity factors, especially for the path-independency of interaction integral. The CAD-based exact representation of tangential and normal vectors facilitates to exactly define a local coordinate system at the crack-tip, and accurately evaluate the crack-face integral. Compared with the standard finite element approach, a higher continuity of stress and strain fields are expected in the interaction integral domain. Various numerical examples of curved crack problems are presented to demonstrate the accuracy of developed isogeometric method through the comparison of both exact and finite element solutions. [ABSTRACT FROM AUTHOR]

An isogeometric shape design sensitivity analysis method is developed for the stress intensity factors (SIFs) in curved crack problems. In order to obtain an enhanced shape sensitivity of SIFs, exact higher-order geometric information of curvature and normal vector is embedded in the isogeometric shape sensitivity expressions. A direct differentiation method is employed for the design sensitivity analysis and the size and orientation of the crack are selected as design variables. In the isogeometric approach, the NURBS basis functions in CAD system are directly utilized in the response analysis, which enables a seamless incorporation of exact geometry and higher continuity into the computational framework. The precise evaluation of the crack-face integral as well as the interaction integral is essential for the computation of the stress intensity factors for the curved crack problems in a mixed-mode. The CAD-based exact representation of tangential and normal vectors enables us to exactly define a local coordinate system at the crack-tip, whose shape dependency naturally leads to configuration design variations that include the change of crack orientation. Compared with the conventional finite element approach, a higher continuity of stress and strain fields are expected in the interaction integral domain. The design dependencies of auxiliary field and q -function are considered in the design sensitivity formulation, reflecting the convection effects due to the non-constant design velocity in the interaction integral domain. Various numerical examples of curved crack problems are presented to verify the developed isogeometric sensitivity analysis method through the comparison with exact and finite element solutions. [ABSTRACT FROM AUTHOR]

Using the isogeometric approach, the variational formulation of both response and sensitivity analyses for geometrically nonlinear structures is derived using the total Lagrangian formulation. The geometric properties of design are embedded in the nonuniform rational B-splines basis functions and control points whose perturbation naturally results in shape changes. Thus, exact geometric models can be utilized in both response and sensitivity analyses. The normal vector and curvature are continuous in the whole design space so that enhanced shape sensitivity can be expected. Refinements and design changes during the shape optimization are easily implemented within the isogeometric framework, which maintains exact geometry without subsequent communication with computer-aided design description. Through numerical examples, the developed isogeometric sensitivity is verified to demonstrate excellent agreement with finite difference sensitivity. Also, the proposed method works very well in various shape optimization problems. [ABSTRACT FROM AUTHOR]

Considering surface effects on nanoscale structures, an isogeometric shape optimization method is developed for curved structures using Naghdi’s shell formulation. Since the curved structures are very sensitive to geometrical changes, the geometric exactness of isogeometric approach successfully prevents the loss of higher-order geometric information in design sensitivity analysis (DSA). A direct differentiation method is employed for the DSA, where the control points are selected as the design variables and describe the flexible modeling of free-form shell surfaces. Through numerical examples, we verified the accuracy of isogeometric analysis whose framework shows better convergence rate than finite element analysis due to the exact geometry and the higher order geometric information in the DSA formulation. The surface Lamé constants turns out to alter the ratio between the portion of membrane and bending energies. Also, the optimal shape is dependent on the residual surface stress that affects its stress state. [ABSTRACT FROM AUTHOR]

An isogeometric topological shape optimization method is developed, using a dual evolution of NURBS curves and level sets; the NURBS curves feature the exact representation of geometry and the level sets help to detect and guide the topological variation of NURBS curves. The implicit geometry by the level sets is transformed into the parametric NURBS curves by minimizing the difference of velocity fields in both representations. A gradient-based optimization problem is formulated, based on the evolution of the NURBS curves. The control points of NURBS curves are taken as design variables. The necessary response and design sensitivity are computed by an isogeometric boundary integral equation method (BIEM) using the NURBS curves. The design sensitivity is obtained on fixed grids and utilized as the velocity to update the Hamilton–Jacobi equation for the level sets. To obtain the whole velocity field on the fixed grids, an interpolation and velocity extension scheme are employed. The developed method provides accurate response and enhanced sensitivity using isogeometric BIEM. Also, additional post-processing is not required to communicate with CAD systems since the optimal design is represented as NURBS curves. Numerical examples demonstrate the accuracy of design sensitivity on fixed grids and the feasibility of shape and topological optimization. [ABSTRACT FROM AUTHOR]

Using an isogeometric approach, a continuum-based configuration design sensitivity analysis (DSA) method is developed for curved Kirchhoff beams with multi-patch junctions. Under the total Lagrangian formulation, large deformations considering the initial curvature of curved beams are described by geometrically exact beam theory (GEBT) and Jaumann strain formulation. In the isogeometric approach, the higher order continuity and the exact description of initial geometry are naturally embedded using NURBS basis functions. In multi-patch models, C 0 -continuity of physical displacement or C 1 -continuity of displacement component at junction is weakly imposed using the Lagrange multiplier method. The superior accuracy of isogeometric analysis (IGA) is verified through the comparison with the results of finite element analysis (FEA) using cubic Hermite interpolation. In the DSA, a material derivative is utilized and the kinematical description of GEBT is consistently employed to express orientation design variations. Contrary to the IGA-based DSA, the Hermite basis function explicitly depends on design in the FEA-based DSA due to its element length parameter. Moreover, since the design velocity field is approximated using the nodal velocity imposed at nodal tangential vector, the amount of design perturbations should be very small to obtain precise design sensitivity. [ABSTRACT FROM AUTHOR]

The shape variation of a domain naturally results in both shape and orientation variations, so called configuration variation, when employing a boundary integral equation (BIE) method. A configuration design sensitivity analysis (DSA) method is developed for steady state heat conduction problems using the boundary integral equations in an isogeometric approach, where NURBS basis functions in a CAD system are directly utilized in the response analysis. Thus, we can accomplish a seamless incorporation of exact geometry and the higher continuity into a computational framework. To enhance the accuracy of configuration design sensitivity, the CAD-based higher-order geometric information such as normal and tangent vectors is exactly embedded in the design sensitivity expressions. The necessary velocity field for configuration design obtained from the NURBS is analytically decomposed into shape and orientation velocity fields. It is shown to be essential to consider orientation variations and significant for accurate configuration sensitivity through comparison with finite differencing conventional BIE method. The developed isogeometric configuration DSA method turns out to be accurate compared with the analytic solution and the conventional DSA method. During the optimization, a mesh regularization scheme is employed to avoid excessive mesh distortion, which comes from significant design changes. [ABSTRACT FROM AUTHOR]

Highlights: [•] NURBS basis functions enable the incorporation of exact geometry into heat conduction analysis. [•] Design parameterization is very easy without subsequent communications with CAD description. [•] Normal vector and curvature are continuous in design space and can enhance the shape sensitivity. [•] The isogeometric adjoint DSA method provides very efficient and accurate shape sensitivity. [•] NURBS basis functions conveniently provide a smooth and non-local design velocity field. [Copyright &y& Elsevier]

Abstract: The isogeometric shape design sensitivity analysis (DSA) includes the desirable features; easy design parameterization and accurate shape sensitivity embedding the higher-order geometric information of curvature and normal vector. Due to the non-interpolatory property of NURBS basis, however, the imposition of essential boundary condition is not so straightforward in the isogeometric method. Taking advantages of geometrically exact property, an isogeometric DSA method is developed applying a mixed transformation to handle the boundary condition. A set of control point and NURBS basis function is added using the h-refinement and Newton iterations to precisely locate the control point to impose the boundary condition. In spite of additional transformation, its computation cost is comparable to the original one with penalty approach since the obtained Kronecker delta property enables to reduce the size of system matrix. Through demonstrative numerical examples, the effectiveness, accuracy, and computing cost of the developed DSA method are discussed. [Copyright &y& Elsevier]

We develop a lattice structure of a desired Poisson's ratio that is maintained when subjected to finite deformation. Using a gradient-based design optimization in an isogeometric computational framework, the representative unit cell is sophisticatedly tailored to attain the controlled Poisson's ratio of up to −2. The ligaments of the lattice structure are modeled using geometrically exact beams whose configuration and cross-sectional area are regarded as design variables, which are parameterized by the higher order B-spline basis functions. The obtained optimal design is fabricated using a PolyJet 3-D printing machine and validated by physical experiments. Excellent quantitative agreement is observed between the numerical and the experimental results measured from an optical deformation measurement system. [ABSTRACT FROM AUTHOR]