Strain Energy Of Structure Research Articles

Incorporating interface effects into multi-material topology optimization by improving interface configuration: An energy-based approach

Interfaces between structural multi-materials generally exhibit asymmetric resistance to tension and compression. Given this interface behavior, this work suggests an energy-based approach to improve the interface configuration for multi-material topology optimization. Based on the strain spectral decomposition, we decompose the structural elastic strain energy into tensile and compressive portions. In the density-based topology optimization framework, we use the gradient-based method to track the interface between multiple materials. Then, we construct an interface-associated scalar field to penalize the tensile portion of the strain energy, causing a pseudo-degradation of the strain energy at the interface region. Finally, within limited material usages and by minimizing the linear weighted structural strain energy and its pseudo-degradation, multi-material topology optimization with improved interface configuration is achieved. Several 2D and 3D numerical examples are investigated, by which the effectiveness and robustness of the suggested approach are fairly validated.

Exploration of the behavior and evolution features of beam-column joints with T-stub using the structural strain energy method

Based on the structural strain energy method, the work behavior of four T-stub connection joints under cyclic loading is analyzed to visualize their working behavior and failure evolution. The experimental strain data are modeled as generalized strain energy density (GSED) to characterize the structural stress state and the corresponding characteristic parameters, and the Mann-Kendall (M-K) criterion is formulated to determine the performance characteristic points (P and Q) of the joint Mj-Ej curves and to derive the starting point of structural continuity damage. Then, by analyzing the parameters of the joint working state (strain, deformation, joint energy dissipation, connection stiffness, bolt force and pry force), the trends of the joint stressing state curves before and after the critical loading are reflected. Finally, the feature point data interpolation (EDI) method is introduced to effectively expand the joint test data, visualize the change process of the strain/stress field, and deeply reveal the mechanical response mechanism and evolutionary degradation law behind it. The contribution of the above work is to quantitatively identify the P and Q characteristic points in the changing trend of the joint working state, to reveal the mechanical mechanism and the failure degradation law from the perspective of energy evolution, and to visualize the mutation of the field data before and after the P/Q using the EDI method, which is a generalized method that can be applied to the other progressive failure structures.

Exploring the Impact of Elements on the Reactivity of a Straightforward Procedure for Generating Vinyl-Carbazole Derivatives via a Frustrated Lewis Pair Mechanism.

The effect of chemical element on the reactivity for carbazolation reaction of phenylacetylene utilizing G13(C6F5)3 (Lewis acid) and G15-carbazole (Lewis base) was theoretically investigated using density functional theory (M06-2X-D3/def2-TZVP), where G13 represents Group 13 elements and G15 represents Group 15 elements. Through activation strain model (ASM) analysis, it is apparent that the reactivity of the entire carbazolation reaction is chiefly governed by the structural strain energy of the alkyne fragment. In other words, if G13(C6F5)3 or G15-carbazole features an atomic radius that is either too small (e.g., B atom) or too large (e.g., Tl or Bi atom), it results in inadequate orbital overlap between the reactants due to the impact of steric effects. This, in turn, results in an elevation of the activation energy for such reactions, thereby impeding the alkyne from undergoing the carbazole catalytic reaction. In light of the above analyses, our theoretical findings suggest that, except for Tl(C6F5)3, the other four Lewis acid catalysts (B(C6F5)3, Al(C6F5)3, Ga((C6F5)3, and In((C6F5)3) demonstrate effectiveness in catalyzing the carbazolation reaction of alkyne alongside with N-carbazole. Additionally, it is anticipated that, among the five categories of G15-carbazole molecules studied, only N-carbazole can participate in the carbazolation reaction with alkyne catalyzed by B(C6F5)3, considering both kinetic and thermodynamic factors at room temperature. Our theoretical investigations, as outlined in this study, indicate that the carbazolation reaction of the alkyne, catalyzed by G13(C6F5)3 and G15-carbazole, follows Hammond's postulate. To put it more plainly, when the transition state of the chemical reaction occurs earlier, it results in a decrease in activation energy.

The influence of invariant horizontal projection area constraint on shape optimization of free-form surface structure with openings

Free-form curved surface shell roof is favored by architects, but the openings for daylighting will decline its mechanical properties. In the shell shape optimization under the vertical load, when the outer contour of the openings or the shell changes, the total load of the structure also changes with the changes in its horizontal projection area. This contradicts the premise of structural optimization, that is, to maintain the principle of unchanging total loads. To solve this problem, the invariant horizontal projection area of the shell was taken as constraints in the shell shape optimization in which the objective function was structural strain energy and optimization design variables were internal and external boundary points of the curved surfaces with openings, and the key points on the curved surface which were design variables of the overall shape of the curved surface. The numerical examples verified the validity of the method, and the influence of the invariant horizontal projection area constraint of the curved surface with openings on the optimization results of the minimum strain energy, optimization with the frequency constraint, and the initial surface with the different support distribution were discussed. The results of the numerical examples showed that the invariant horizontal projection area constraint has a greater impact on the results of the curved surface with opening optimization, and its influence should be considered in the engineering design.

Morphogenesis of free-form surface structures based on geometrically nonlinear isogeometric analysis

Free-form surface structures have taken the forefront in space structure research owing to their free and flexible expressiveness and aesthetic visual effects. Yet, their irregular shapes and large span of the initial structure often lead to the problem of large displacement, necessitating the incorporation of geometric nonlinearity in the analysis. In this paper, we employ structural optimization design principles to investigate the morphogenesis of free-form surface structures while considering their geometric nonlinearity. Particularly, we explore both single-objective and double-objective shape optimization of free-form surface structures by employing structural strain energy and the first natural frequency as optimization indexes and the vertical coordinates of control points as optimization variables. The effectiveness of the proposed method is finally verified by several examples.

Multi-objective shape-section optimization of free-form latticed shells using the RBF-NSGA-II algorithm

In this study, a novel RBF-NSGA-II optimization framework based on the response surface methodology (RSM) and the constrained NSGA-II algorithm is proposed to simultaneously improve the static, nonlinear stability and economic performance of free-form latticed shells. The bending stiffness of joints is innovatively considered one of the design variables for shape-member section coupled optimization. The augmented radius basis functions (RBFs) are adopted to develop the metamodels of both the objectives and constraints. The improved minimum distance selection method (TMDSM) is used to select the optimal solution from the Pareto optimal solution set. Moreover, three alternative solutions are presented as supplements to meet various engineering requirements. The proposed RBF-NSGA-II algorithm is validated to have high accuracy for multi-objective optimization problems with high nonlinearity. Compared with those of the initial structures, the strain energy and steel consumption of the optimal structure decreased by 82.23 % and 17.97 %, respectively, while the buckling load considering both the geometric nonlinearity and material nonlinearity increased by 1.2 times. The involvement of the joint stiffness is demonstrated to have a great influence on the multi-objective optimization results. It is necessary to consider the joint stiffness as one of the design variables during shape-member section coupled optimization.

ЭНЕРГЕТИЧЕСКИЕ СВОЙСТВА СИММЕТРИЧНЫХ ДЕФОРМИРУЕМЫХ СИСТЕМ

Energy methods for calculating structures, which have become popular for a century, are based on the Lagrange principle and have the meaning of equality of work of external forces and internal forces. Having proved their effectiveness in the overwhelming majority of problems of structural mechanics, they became the dominant approach in formulating the problems of studying solid deformable systems and gave rise to the main methodology for solving problems. As a result, a situation has arisen that the internal potential energy of a deformed body remains insufficiently studied.
 The paper develops an approach to the study of the symmetric structure at critical levels of strain energy. The criterion of critical levels of strain energy, based on the concepts of "self-stress" ("self-balance") of a deformable body. Limiting values of the structure strain energy may get by varying the reactions and deflections in the nodal points. The extreme values of forces and displacements of the rods are calculated in matrix form from the values of nodal reactions (displacements).
 Methodology for studying the energy properties of a system is shown on the examples of the study of symmetric rod systems without involving the concept of external forces. The technique is based on matrix methods of structural mechanics and the mathematical apparatus of eigenvalue problems. The comparison of structural design and structural analysis solution of structural mechanics tasks by traditional methods and with the proposed methodology is carried out.

Structural overall damage index based on structural strain energy

In order to accurately evaluate the overall damage degree of a structure after an earthquake and provide guidance for structural reinforcement and repair, it is crucial to develop a reasonable damage model that can effectively describe the structural overall damage. This has been a focal point of research in the field of seismic engineering. Structural strain energy (SSE) is a parameter that can sensitively reflect the damage condition of the entire structure. Therefore, based on SSE and considering the coupling effect of the first transcendental failure and cumulative hysteretic energy dissipation, a structural overall damage index is derived through theoretical analysis. This index aims to capture the extent of damage based on SSE. To validate the credibility of the proposed damage index, a 60-story truss frame-core tube structure model with outrigger (TCO), measuring 237.60 m in height, is selected as an example. Six representative damage indices are chosen for comparison purposes. The nonlinear dynamic characteristics of the entire structure are analyzed using incremental dynamic analysis (IDA). Additionally, the seven damage indices, including the code-specified values, are compared to verify the credibility and applicability of the proposed structural overall damage index.

Isogeometric double-objective shape optimization of free-form surface structures with Kirchhoff–Love shell theory

Free-form surface structures are extensively utilized in practical engineering due to their rich architectural expression and strong visual impact. However, ensuring good mechanical behavior during the generation process of free-form surface structures has become a critical problem in the scientific community. Isogeometric analysis (IGA) is a widely adopted approach in various fields owing to its numerous advantages, such as geometric accuracy, high-order continuity, high precision, without traditional meshing, and ease of integration with CAD. The presented work adopts the NURBS surfaces to create geometric models, and developes an isogeometric discretization for free-form surfaces by formulating a shell analysis method based on Kirchhoff–Love hypothesis. The structural strain energy and the first natural frequency of the free-form surfaces are selected as the static and dynamic performance evaluation indices. The optimization objectives are to minimize the structural strain energy and maximize the first natural frequency, with the control point coordinates considered as the optimization variable. The double-objective shape optimization of the free-form surface structure is carried out using the particle swarm optimization technique. The effectiveness and performance of the proposed method are verified through five numerical examples, and the mechanical properties of the optimized structure are substantially improved.

Level set function based on conformal geometry theory: An efficient approach for optimization of shell structure with arbitrary Gaussian curvature

Topology optimization based on the level set (LS) theory is of great significance to shell structure design. However, the Gaussian curvature of the shell structure can turn complex in many cases, and the establishing process of the LS function will become extremely cumbersome, which consumes overmuch time to solve the Hamilton-Jacobi(H-J) equation. In order to improve the efficiency of optimization, an LS function based on the conformal geometry theory (LSF-CGT) is proposed in this paper. Firstly, the shell structure with arbitrary Gaussian curvature is conformally mapped into 2D Euclidean space to generate the orthogonal mesh of the preset initial LS function. Then the barycentric interpolation method is used to reveal the transformation relationship between the orthogonal and shell structure mesh. Finally, the LS function of the orthogonal mesh is assigned to the mapped shell structure, and the LS function of the 3D shell structure is subsequently obtained. Three typical shell structures with positive, zero, and negative Gaussian curvature are used to verify the effectiveness of LSF-CGT in topology optimization. Results show that LSF-CGT can effectively reduce the strain energy of shell structures with arbitrary Gaussian curvatures by more than 30%. As a method for optimizing shell structures with arbitrary Gaussian curvatures, LSF-CGT can not only stably and quickly converge to the optimal structure in the 2D domain but also effectively maintain the initial geometric characteristics of the shell structure.

Instability-Induced Origami Design by Topology Optimization

Instability-induced wrinkle patterns of thin sheets are ubiquitous in nature, which often result in origami-like patterns that provide inspiration for the engineering of origami designs. Inspired by instability-induced origami patterns, we propose a computational origami design method based on the nonlinear analysis of loaded thin sheets and topology optimization. The bar-and-hinge model is employed for the nonlinear structural analysis, added with a displacement perturbation strategy to initiate out-of-plane buckling. Borrowing ideas from topology optimization, a continuous crease indicator is introduced as the design variable to indicate the state of a crease, which is penalized by power functions to establish the mapping relationships between the crease indicator and hinge properties. Minimizing the structural strain energy with a crease length constraint, we are able to evolve a thin sheet into an origami structure with an optimized crease pattern. Two examples with different initial setups are illustrated, demonstrating the effectiveness and feasibility of the method.

Interbody Fusion Cage Design Driven by Topology Optimization

We used topology optimization technology to explore the new theory and method of interbody fusion cage design and realized an innovative design of interbody cages. The lumbar spine of a normal healthy volunteer was scanned to perform reverse modeling. Based on the scan data for the L1-L2 segments of the lumbar spine, a three dimensional model was reconstructed to obtain the complete simulation model of the L1-L2 segment. The boundary inversion method was used to obtain approximately isotropic material parameters that can effectively characterize the mechanical behavior of vertebrae, thereby reducing the computational complexity. The topology description function was used to model the clinically used traditional fusion cage to obtain Cage A. The moving morphable void-based topology optimization method was used for the integrated design of size, shape, and topology to obtain the optimized fusion cage, Cage B. The volume fraction of the bone graft window in Cage B was 74.02%, which was 60.67% higher than that (46.07%) in Cage A. Additionally, the structural strain energy in the design domain of Cage B was 1.48mJ, which was lower than that of Cage A (satisfying the constraints). The maximum stress in the design domain of Cage B was 5.336 Mpa, which was 35.6% lower than that (8.286 Mpa) of Cage A. In addition, the surface stress distribution of Cage B was more uniform than that of Cage A. This study proposed a new innovative design method for interbody fusion cages, which not only provides new insights into the innovative design of interbody fusion cages but may also guide the customized design of interbody fusion cages in different pathological environments.

A general theory for analyzing morphing bistable tensegrities based on quasi-static assumption

Based on the energy method and quasi-static assumption, a general theory in matrix form is developed to analyze the kinematics of morphing bistable tensegrities and the transfer between paths. The tangent stiffness matrix of a tensegrity is deduced based on the variation of the potential energy of the system, and an analytical expression of its partial derivative with respect to the node coordinates is also provided. A strategy for analyzing the kinematic path of tensegrities under member length actuation is proposed. It is proven that the two kinematic paths, which are in the manifold and under the same member length actuation, have no intersection. Based on the relationship between the structural deformation and the elastic strain energy of a tensegrity, an approach to trace the transfer branch between bistable configurations on different kinematic paths is proposed considering both the target direction toward the other bistable configuration and the increment of the structural elastic strain energy. A strategy for crossing over the snapping instability zone on the transfer branch is suggested. The difficulty of the transfer can thus be evaluated by the depth of the potential well. Two illustrative examples are employed to verify the validity of the proposed method by analyzing their kinematic paths and the transfer branches.

Nonlinear vibration of electro-rheological sandwich plates, coupled to quiescent fluid

The current paper aims to study the free vibration analysis of a sandwich plate in contact with quiescent fluid. The structure is made of an electro-rheological (ER) fluid core, and two laminated composite skins. The fluid is ideal, interacting partially with a plate, and fluid potential velocity is derived, satisfying continuity equation. Higher-order shear deformable theory is employed for the displacement field, and nonlinear Von-Karman strains are taken into account. After calculating the structural strain energy and kinetic energies of the fluid and plate, Galerkin approach and harmonic balance method are adopted to obtain natural frequencies of the system. Next, the accuracy and reliability of the developed model are confirmed, and then a detailed parametric study is conducted to examine the effects of different parameters such as geometrical dimensions of plate, fluid parameters and electric field strength on vibrational characteristics of the structure. The novelty of the presented model focuses on the nonlinear analysis of an ER sandwich panel coupled to ideal fluid, and the obtained results can guide the future design of composite ER panels.

Topology Optimization of Stiffener Layout Design for Box Type Load-Bearing Component under Thermo-Mechanical Coupling

The structure optimization design under thermo-mechanical coupling is a difficult problem in the topology optimization field. An adaptive growth algorithm has become a more effective approach for structural topology optimization. This paper proposed a topology optimization method by an adaptive growth algorithm for the stiffener layout design of box type load-bearing components under thermo-mechanical coupling. Based on the stiffness diffusion theory, both the load stiffness matrix and the heat conduction stiffness matrix of the stiffener are spread at the same time to make sure the stiffener grows freely and obtain an optimal stiffener layout design. Meanwhile, the objectives of optimization are the minimization of strain energy and thermal compliance of the whole structure, and thermo-mechanical coupling is considered. Numerical studies for square shells clearly show the effectiveness of the proposed method for stiffener layout optimization under thermo-mechanical coupling. Finally, the method is applied to optimize the stiffener layout of box type load-bearing component of the machining center. The optimization results show that both the structural deformation and temperature of the load-bearing component with the growth stiffener layout, which are optimized by the adaptive growth algorithm, are less than the stiffener layout of shape ‘#’ stiffener layout. It provides a new solution approach for stiffener layout optimization design of box type load-bearing components under thermo-mechanical coupling.

Structural Damage Detection Using Convolutional Neural Networks Based on Modal Strain Energy and Population of Structures

A convolutional neural network (CNN)-based structural damage detection (SDD) method using populations of structures and modal strain energy (MSE) is proposed. In this study, sufficient samples of the CNN are provided by numerical simulations, and the size of the model can be changed by modifying the coordinates of some nodes, thereby establishing a series of numerical models (i.e., a population). Finally, three groups are investigated, the effects of multiple indices on damage detection based on population are compared. The results demonstrate that the MSE as a damage index is superior to the other indices.

Isogeometric Level Set-Based Topology Optimization for Geometrically Nonlinear Plane Stress Problems

This paper addresses geometrically nonlinear topology optimization using IsoGeometric Analysis (IGA) and Level Set Method (LSM) under plane stress assumptions. The IGA is employed to solve governing equations and accurately describe geometric modeling. The pointwise gradient-based sensitivity analysis is applied to derive the normal velocity of the Reaction–Diffusion Equation (RDE) to update the control net of the Level Set Function (LSF). The Generalized Displacement Control Method (GDCM) is used to obtain the equilibrium path based on the Total Lagrange (TL) formulation. A pointwise energy interpolation strategy is employed to avoid low-density instabilities. The exact Heaviside function is also applied to obtain a 0/1 manufacturable design. The objective is to maximize the total strain energy of structure, which is defined as the total external work within a specified displacement, while a certain volume of the design domain is considered. To demonstrate the ability and efficiency of the proposed method, several numerical examples are presented. By making comparisons, the results of both optimization algorithm and nonlinear analysis are shown to be in good agreement with the literature. Also, it is shown that obtained layouts are different from those in the linear behavior modeling due to considering buckling effects. In addition, linear and nonlinear final layouts are analyzed under large deformation assumption and the load–displacement curves are compared to illustrate the improvement of the objective function and final load. • Optimized topologies are parameterized using NURBS. • A pointwise energy interpolation strategy is employed to avoid low-density instabilities. • The control net of the design domain is updated by the reaction–diffusion equation. • Pointwise sensitivity analysis of the optimization problem is analytically derived. • 0/1 manufacturable design was performed using an exact (binary) Heaviside step function.

Umbrella meshes

We present a computational inverse design framework for a new class of volumetric deployable structures that have compact rest states and deploy into bending-active 3D target surfaces. Umbrella meshes consist of elastic beams, rigid plates, and hinge joints that can be directly printed or assembled in a zero-energy fabrication state. During deployment, as the elastic beams of varying heights rotate from vertical to horizontal configurations, the entire structure transforms from a compact block into a target curved surface. Umbrella Meshes encode both intrinsic and extrinsic curvature of the target surface and in principle are free from the area expansion ratio bounds of past auxetic material systems. We build a reduced physics-based simulation framework to accurately and efficiently model the complex interaction between the elastically deforming components. To determine the mesh topology and optimal shape parameters for approximating a given target surface, we propose an inverse design optimization algorithm initialized with conformal flattening. Our algorithm minimizes the structure's strain energy in its deployed state and optimizes actuation forces so that the final deployed structure is in stable equilibrium close to the desired surface with few or no external constraints. We validate our approach by fabricating a series of physical models at various scales using different manufacturing techniques.

Variable-thickness optimization method for shell structures based on a regional evolutionary control strategy

Shell structures are widely used in various engineering applications due to their efficient load-carrying capacity relative to the material volume. However, the degree of freedom of most structural optimizations is too large to control the overall thickness of shell structures. To address this issue, this paper proposes a variable-thickness structural optimization method for shell structures based on a regional evolutionary control strategy to reduce the overall structural strain energy and volume. The essential object of our method is to define the thickness element as the design variable, which can constrain the external surface of the shell structure and control the thickness variation of its internal surface in the local area. Then, we introduce a regional evolutionary control strategy that can adaptively obtain and update the evolutionary hierarchy centers and dynamically update the evolutionary factors. By alternately performing factor evolutionary optimization and regional evolutionary optimization to minimize the strain energy and volume of shell structures. We demonstrate that variable-thickness shell structures can achieve good static performance and explore the trade-off between the strain energy and volume.

Mechanism of Hydraulic Fracturing Cutting Hard Basic Roof to Prevent Rockburst

Rockburst is globally regarded as one of the most severe and complicated mining dynamic disasters to predict or control. Generally, the occurrence mechanism of rockbursts can be considered as a process of the elastic strain energy accumulation, emancipation, transmission, and occurrence. Tracing to the source, the reasons for large accumulation of elastic strain energy in coal and rock mass are the high stress of the roof layer that loads on the coal and rock masses around the mining space coupling effect with the natural horizontal tectonic stress. In this study, using the minimum energy theory and elasticity theory, the analytical formula for calculating elastic strain energy of the roof cantilever beam structure acting on the coal body load in front of the working face is deduced. Accordingly, we achieved a method of using hydraulic fracturing to improve the roof structure. In detail, we use a high-pressure jet to cut the cantilever roof structure, which can make a prelocated fracture surface, and then utilize the packers to make sure that the injected high-pressure fracturing fluid is propagating along the prelocated fracture surface and can cut off the cantilever roof structure eventually to prevent rockbursts in advance. Due to the rockburst occurrence mechanism and the quantitatively elastic strain energy analytical formula, a preconditioning water jet cutting induced fracture surface to create orientation-controllable hydraulic fracture strategy is proposed to guard against the high hazard caused by the massive elastic strain energy, which accumulated in the coal body in front of the working face and coal pillar.