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"Structural optimization Mathematics."
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Optimization and anti-optimization of structures under uncertainty
The volume presents a collaboration between internationally recognized experts on anti-optimization and structural optimization, and summarizes various novel ideas, methodologies and results studied over 20 years. The book vividly demonstrates how the concept of uncertainty should be incorporated in a rigorous manner during the process of designing real-world structures. The necessity of anti-optimization approach is first demonstrated, then the anti-optimization techniques are applied to static, dynamic and buckling problems, thus covering the broadest possible set of applications. Finally, anti-optimization is fully utilized by a combination of structural optimization to produce the optimal design considering the worst-case scenario. This is currently the only book that covers the combination of optimization and anti-optimization. It shows how various optimization techniques are used in the novel anti-optimization technique, and how the structural optimization can be exponentially enhanced by incorporating the concept of worst-case scenario, thereby increasing the safety of the structures designed in various fields of engineering.
eBook
Gradient methods for minimizing composite functions
In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two terms: one is smooth and given by a black-box oracle, and another is a simple general convex function with known structure. Despite the absence of good properties of the sum, such problems, both in convex and nonconvex cases, can be solved with efficiency typical for the first part of the objective. For convex problems of the above structure, we consider primal and dual variants of the gradient method (with convergence rate
), and an accelerated multistep version with convergence rate
, where
is the iteration counter. For nonconvex problems with this structure, we prove convergence to a point from which there is no descent direction. In contrast, we show that for general nonsmooth, nonconvex problems, even resolving the question of whether a descent direction exists from a point is NP-hard. For all methods, we suggest some efficient “line search” procedures and show that the additional computational work necessary for estimating the unknown problem class parameters can only multiply the complexity of each iteration by a small constant factor. We present also the results of preliminary computational experiments, which confirm the superiority of the accelerated scheme.
Journal Article
Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics
Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.
Journal Article
Regularized and Structured Tensor Total Least Squares Methods with Applications
Total least squares (TLS), also named as errors in variables in statistical analysis, is an effective method for solving linear equations with the situations, when noise is not just in observation data but also in mapping operations. Besides, the Tikhonov regularization is widely considered in plenty of ill-posed problems. Moreover, the structure of mapping operator plays a crucial role in solving the TLS problem. Tensor operators have some advantages over the characterization of models, which requires us to build the corresponding theory on the tensor TLS. This paper proposes tensor regularized TLS and structured tensor TLS methods for solving ill-conditioned and structured tensor equations, respectively, adopting a tensor-tensor-product. Properties and algorithms for the solution of these approaches are also presented and proved. Based on this method, some applications in image and video deblurring are explored. Numerical examples illustrate the effectiveness of our methods, compared with some existing methods.
Journal Article
Existence and Stability of Fuzzy Slightly Altruistic Equilibrium for a Class of Generalized Multiobjective Fuzzy Games
We mainly study the existence, structural stability and robustness of fuzzy slightly altruistic equilibria for a class of generalized multiobjective fuzzy games which are expressed as ϖ. Firstly, we introduce the concept of fuzzy slightly altruistic equilibrium and prove the existence of equilibrium for the ϖ by Fan–Glicksberg fixed point theorem. Secondly, the connections between ϖ and bounded rationality are discussed by an abstract rationality functions. Moreover, we construct the problem space of ϖ which is represented by Λ and show that most ϖ∈Λ are structurally stable and robust to ε-equilibrium on the meaning of Baire category. These results are new and extend some existing results in recent literature.
Journal Article
Topology, Size, and Shape Optimization in Civil Engineering Structures: A Review
The optimization of civil engineering structures is critical for enhancing structural performance and material efficiency in engineering applications. Structural optimization approaches seek to determine the optimal design, by considering material performance, cost, and structural safety. The design approaches aim to reduce the built environment’s energy use and carbon emissions. This comprehensive review examines optimization techniques, including size, shape, topology, and multi-objective approaches, by integrating these methodologies. The trends and advancements that contribute to developing more efficient, cost-effective, and reliable structural designs were identified. The review also discusses emerging technologies, such as machine learning applications with different optimization techniques. Optimization of truss, frame, tensegrity, reinforced concrete, origami, pantographic, and adaptive structures are covered and discussed. Optimization techniques are explained, including metaheuristics, genetic algorithm, particle swarm, ant-colony, harmony search algorithm, and their applications with mentioned structure types. Linear and non-linear structures, including geometric and material nonlinearity, are distinguished. The role of optimization in active structures, structural design, seismic design, form-finding, and structural control is taken into account, and the most recent techniques and advancements are mentioned.
Journal Article
Structural Displacement Requirement in a Topology Optimization Algorithm Based on Isogeometric Entities
This work deals with the formulation of a general design requirement on the displacement of a continuum medium in the framework of a special density-based algorithm for topology optimization. The algorithm makes use of non-uniform rational basis spline hyper-surfaces to represent the pseudo-density field describing the part topology and of the well-known solid isotropic material with penalization approach. The proposed formulation efficiently exploits the properties of the isogeometric basis functions, which allow defining an implicit filter. In particular, the structural displacement requirement is formulated in the most general sense, by considering displacements on loaded and non-loaded regions. The gradient of the structural displacement is evaluated in closed form by using the principle of virtual work. Moreover, a sensitivity analysis of the optimized topology to the integer parameters, involved in the definition of the hyper-surface, is carried out. The effectiveness of the proposed approach is proven through meaningful 2D and 3D benchmarks.
Journal Article
Structural Design Optimization Using Isogeometric Analysis: A Comprehensive Review
Isogeometric analysis (IGA), an approach that integrates CAE into conventional CAD design tools, has been used in structural optimization for 10 years, with plenty of excellent research results. This paper provides a comprehensive review on isogeometric shape and topology optimization,
with a brief coverage of size optimization. For isogeometric shape optimization, attention is focused on the parametrization methods, mesh updating schemes and shape sensitivity analyses. Some interesting observations, e.g. the popularity of using direct (differential) method for shape sensitivity
analysis and the possibility of developing a large scale, seamlessly integrated analysis-design platform, are discussed in the framework of isogeometric shape optimization. For isogeometric topology optimization (ITO), we discuss different types of ITOs, e.g. ITO using SIMP (Solid Isotropic
Material with Penalization) method, ITO using level set method, ITO using moving morphable components (MMC), ITO with phase field model, etc.,their technical details and applications such as the spline filter, multi-resolution approach, multi-material problems and stress constrained problems.
In addition to the review in the last 10 years, the current developmental trend of isogeometric structural optimization is discussed.
Journal Article
Long-Time Behavior of a Nonlinearly-Damped Three-Layer Rao–Nakra Sandwich Beam
In this paper, a three-layer Rao–Nakra sandwich beam is considered where the core viscoelastic layer is constrained by the purely elastic or piezoelectric outer layers. In the model, uniform bending motions of the overall laminate are coupled to the longitudinal motions of the outer layers, and the shear of the middle layer contributes to the overall motion. Together with nonlinear damping injection and nonlinear source terms, the existence and uniqueness of local and global weak solutions are obtained by the nonlinear semigroup theory and the theory of monotone operators. The global existence of potential well solutions and the uniform energy decay rates of such a solution, given as a solution to a certain nonlinear ODE, are shown are proved under certain assumptions of the parameters and by the Nehari manifold. Finally, the existence of a smooth global attractor with finite fractal dimension, which is characterized as an unstable manifold of the set of stationary solutions, and exponential attractors for the associated dynamical system are proved. The present paper extends the linear analysis of the stability of the Rao–Nakra sandwich beam to nonlinear analysis in the existing literature.
Journal Article