Modeling Geometrically Nonlinear FG Plates: A Fast and Accurate Alternative to IGA Method Based on Deep Learning

Isogeometric analysis (IGA) is known to showadvanced features compared to traditional finite element approaches.Using IGA one may accurately obtain the geometrically nonlinear bending behavior of plates with functionalgrading (FG). However, the procedure is usually complex and often is time-consuming. We thus put forward adeep learning method to model the geometrically nonlinear bending behavior of FG plates, bypassing the complexIGA simulation process. A long bidirectional short-term memory (BLSTM) recurrent neural network is trainedusing the load and gradient index as inputs and the displacement responses as outputs. The nonlinear relationshipbetween the outputs and the inputs is constructed usingmachine learning so that the displacements can be directlyestimated by the deep learning network. To provide enough training data, we use S-FSDT Von-Karman IGA andobtain the displacement responses for different loads and gradient indexes. Results show that the recognition erroris low, and demonstrate the feasibility of deep learning technique as a fast and accurate alternative to IGA formodeling the geometrically nonlinear bending behavior of FG plates.

Geometrically Nonlinear Flutter Analysis Based on CFD/CSD Methods and Wind Tunnel Experimental Verification

This study presents a high-speed geometrically nonlinear flutter analysis calculation method based on the highprecision computational fluid dynamics/computational structural dynamics methods.In the proposed method,the aerodynamic simulation was conducted based on computational fluid dynamics,and the structural model was established using the nonlinear finite element model and tangential stiffness matrix.First,the equilibrium position was obtained using the nonlinear static aeroelastic iteration.Second,the structural modal under a steady aerodynamic load was extracted.Finally,the generalized displacement time curve was obtained by coupling the unsteady aerodynamics and linearized structure motion equations.Moreover,if the flutter is not at a critical state,the incoming flow dynamic pressure needs to be changed,and the above steps must be repeated until the vibration amplitude are equal.Furthermore,the high-speed geometrically nonlinear flutter of the wing-body assemblymodel with a high-aspect ratio was investigated,and the correctness of the method was verified using high-speed wind tunnel experiments.The results showed that the geometric nonlinearity of the large deformation of the wing caused in-plane bending to become a key factor in flutter characteristics and significantly decreased the dynamic pressure and frequency of the nonlinear flutter compared to those of the linear flutter.

GEOMETRICALLY NONLINEAR FINITE ELEMENT MODEL OF SPATIAL THIN-WALLED BEAMS WITH GENERAL OPEN CROSS SECTION

Based on the theory of Timoshenko and thin-walled beams, a new finite element model of spatial thin-walled beams with general open cross sections is presented in the paper, in which several factors are included such as lateral shear deformation, warp generated by nonuni- form torsion and second-order shear stress, coupling of flexure and torsion, and large displacement with small strain. With an additional internal node in the element, the element stiffness matrix is deduced by incremental virtual work in updated Lagrangian （UL） formulation. Numerical examples demonstrate that the presented model well describes the geometrically nonlinear property of spatial thin-walled beams.

A MESHLESS LOCAL PETROV-GALERKIN METHOD FOR GEOMETRICALLY NONLINEAR PROBLEMS

Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.

New tangent stiffness matrix for geometrically nonlinear analysis of space frames

A three-dimensional beam element is derived based on the principle of stationary total potential energy for geometrically nonlinear analysis of space frames. A new tangent stiffness matrix, which allows for high order effects of element deformations, replaces the conventional incremental secant stiffness matrix. Two deformation stiffness matrices due to the variation of axial force and bending moments are included in the tangent stiffness. They are functions of element deformations and incorporate the coupling among axial, lateral and torsional deformations. A correction matrix is added to the tangent stiffness matrix to make displacement derivatives equivalent to the commutative rotational degrees of freedom. Numerical examples show that the proposed dement is accurate and efficient in predicting the nonlinear behavior, such as axial-torsional and flexural-torsional buckling, of space frames even when fewer elements are used to model a member.

Geometrically Nonlinear Analysis of Structures Using Various Higher Order Solution Methods: A Comparative Analysis for Large Deformation

The suitability of six higher order root solvers is examined for solving the nonlinear equilibrium equations in large deformation analysis of structures.The applied methods have a better convergence rate than the quadratic Newton-Raphson method.These six methods do not require higher order derivatives to achieve a higher convergence rate.Six algorithms are developed to use the higher order methods in place of the Newton-Raphson method to solve the nonlinear equilibrium equations in geometrically nonlinear analysis of structures.The higher order methods are applied to both continuum and discrete problems(spherical shell and dome truss).The computational cost and the sensitivity of the higher order solution methods and the Newton-Raphson method with respect to the load increment size are comparatively investigated.The numerical results reveal that the higher order methods require a lower number of iterations that the Newton-Raphson method to converge.It is also shown that these methods are less sensitive to the variation of the load increment size.As it is indicated in numerical results,the average residual reduces in a lower number of iterations by the application of the higher order methods in the nonlinear analysis of structures.

GEOMETRICALLY NONLINEAR ANALYSIS OF MINDLIN PLATEUSING THE INCOMPATIBLE BENDING ELEMENTSWITH INTERNAL SHEAR STRAIN

An approach of the incompatible elements with additional internal shear strain is,in the presem paper,suggested and applied to geometrically nonlinear analysis of Mi-ndlin plate bending problem.It provides a quite covenient way to avoid the whear loc-king troubles.An energy consistency condition for this kind of C°elements is offered.The nonlinear element formulations and some numerical results are presented.

TOPOLOGY SYNTHESIS OF GEOMETRICALLY NONLINEAR COMPLIANT MECHANISMS USING MESHLESS METHODS

This paper presents a new method for topology optimization of geometrical nonlinear compliant mechanisms using the element-free Galerkin method （EFGM）. The EFGM is employed as an alternative scheme to numerically solve the state equations by fully taking advantage of its capability in dealing with large displacement problems. In the meshless method, the imposition of essential boundary conditions is also addressed. The popularly studied solid isotropic material with the penalization （SIMP） scheme is used to represent the nonlinear dependence between material properties and regularized discrete densities. The output displacement is regarded as the objective function and the adjoint method is applied to finding the sensitivity of the design functions. As a result, the optimization of compliant mechanisms is mathematically established as a nonlinear programming problem, to which the method of moving asymptotes （MMA） belonging to the sequential convex programming can be applied. The availability of the present method is finally demonstrated with several widely investigated numerical examples.

Geometrically Nonlinear Random Responses of Stiffened Plates Under Acoustic Pressure

An algorithm integrating reduced order model(ROM),equivalent linearization(EL),and finite element method(FEM)is proposed to carry out geometrically nonlinear random vibration analysis of stiffened plates under acoustic pressure loading.Based on large deflection finite element formulation,the nonlinear equations of motion of stiffened plates are obtained.To reduce the computation,a reduced order model of the structures is established.Then the EL technique is incorporated into FE software NASTRAN by the direct matrix abstraction program(DMAP).For the stiffened plates,a finite element model of beam and plate assembly is established,in which the nodes of beam elements are shared with shell elements,and the offset and section properties of the beam are set.The presented method can capture the root-mean-square(RMS) of the stress responses of shell and beam elements of stiffened plates,and analyze the stress distribution of the stiffened surface and the unstiffened surface,respectively.Finally,the statistical dynamic response results obtained by linear and EL methods are compared.It is shown that the proposed method can be used to analyze the geometrically nonlinear random responses of stiffened plates.The geometric nonlinearity plays an important role in the vibration response of stiffened plates,particularly at high acoustic pressure loading.

Topology Optimization of Geometrically Nonlinear Structures Under Thermal-Mechanical Coupling

A geometrically nonlinear topology optimization(GNTO)method with thermal–mechanical coupling is investigated.Firstly,the new expression of element coupling stress due to superimposed mechanical and thermal loading is obtained based on the geometrically nonlinear finite element analysis.The lightweight topology optimization(TO)model under stress constraints is established to satisfy the strength requirement.Secondly,the distortion energy theory is introduced to transform themodel into structural strain energy constraints in order to solve the implicit relationship between stress constraints and design variables.Thirdly,the sensitivity analysis of the optimization model is derived,and the model is solved by the method of moving asymptotes(MMA).Numerical examples show that temperature has a significant effect on the optimal configuration,and the TO method considering temperature load is closer to engineering design requirements.The proposed method can be extended to the GNTO design with multiple physical field coupling.

Geometrically Nonlinear Deformation Reconstruction Based on iQS4 Elements Using a Linearized Iterative iFEM Algorithm

Structural shape monitoring plays a vital role in the structural health monitoring systems.The inverse finite element method(iFEM)has been demonstrated to be a practical method of deformation reconstruction owing to its unique advantages.Current iFEM formulations have been applied to small deformation of structures based on the small-displacement assumption of linear theory.However,this assumption may be inapplicable to some structures with large displacements in practical applications.Therefore,geometric nonlinearity needs to be considered.In this study,to expand the practical utility of iFEM for large displacement monitoring,we propose a nonlinear iFEM algorithm based on a four-node inverse quadrilateral shell element iQS4.Taking the advantage of an iterative iFEM algorithm,a nonlinear response is linearized to compute the geometrically nonlinear deformation reconstruction,like the basic concept of nonlinear FE analysis.Several examples are solved to verify the proposed approach.It is demonstrated that large displacements can be accurately estimated even if the in-situ sensor data includes different levels of randomly generated noise.It is proven that the nonlinear iFEM algorithm provides a more accurate displacement response as compared to the linear iFEM methodology for structures undergoing large displacement.Hence,the proposed approach can be utilized as a viable tool to effectively characterize geometrically nonlinear deformations of structures in real-time applications.

Numerical Exploration of Asymmetrical Impact Dynamics: Unveiling Nonlinearities in Collision Problems and Resilience of Reinforced Concrete Structures

This study provides a comprehensive analysis of collision and impact problems’ numerical solutions, focusing ongeometric, contact, and material nonlinearities, all essential in solving large deformation problems during a collision.The initial discussion revolves around the stress and strain of large deformation during a collision, followedby explanations of the fundamental finite element solution method for addressing such issues. The hourglassmode’s control methods, such as single-point reduced integration and contact-collision algorithms are detailedand implemented within the finite element framework. The paper further investigates the dynamic responseand failure modes of Reinforced Concrete (RC) members under asymmetrical impact using a 3D discrete modelin ABAQUS that treats steel bars and concrete connections as bond slips. The model’s validity was confirmedthrough comparisons with the node-sharing algorithm and system energy relations. Experimental parameterswere varied, including the rigid hammer’s mass and initial velocity, concrete strength, and longitudinal and stirrupreinforcement ratios. Findings indicated that increased hammer mass and velocity escalated RC member damage,while increased reinforcement ratios improved impact resistance. Contrarily, increased concrete strength did notsignificantly reduce lateral displacement when considering strain rate effects. The study also explores materialnonlinearity, examining different materials’ responses to collision-induced forces and stresses, demonstratedthrough an elastic rod impact case study. The paper proposes a damage criterion based on the residual axialload-bearing capacity for assessing damage under the asymmetrical impact, showing a correlation betweendamage degree hammer mass and initial velocity. The results, validated through comparison with theoreticaland analytical solutions, verify the ABAQUS program’s accuracy and reliability in analyzing impact problems,offering valuable insights into collision and impact problems’ nonlinearities and practical strategies for enhancingRC structures’ resilience under dynamic stress.

Modified unified co-rotational framework with beam,shell and brick elements for geometrically nonlinear analysis

The co-rotational finite element formulation is an attractive technique extending the capabilities of an existing high performing linear element to geometrically nonlinear analysis.This paper presents a modified co-rotational framework,unified for beam,shell,and brick elements.A unified zero-spin criterion is proposed to specify the local element frame,whose origin is always located at the centroid.Utilizing this criterion,a spin matrix is introduced,and the local frame is invariant to the element nodal ordering.Additionally,the projector matrix is redefined in a more intuitive way,which is the derivative of local co-rotational element frame with respect to the global one.Furthermore,the nodal rotation is obtained with pseudo vector and instantaneous rotation,under a high-order accurate transformation.The resulting formulations are achieved in unified expression and thus a series of linear elements can be embedded into the framework.Several examples are presented to demonstrate the efficiency and accuracy of the proposed framework for large displacement analysis.

LINEAR AND GEOMETRICALLY NONLINEAR ANALYSIS WITH 4-NODE PLANE QUASI-CONFORMING ELEMENT WITH INTERNAL PARAMETERS

A linear 4-node quadrilateral quasi-conforming plane element with internal parameters is proposed. The element preserves advantages of the quasi-conforming technique, including an explicit stiffness matrix, which can be applied to nonlinear problems. The weak patch test guarantees the convergence of the element. Then the linear element is extended to the geometri- cally nonlinear analysis in the framework of Total Lagrangian （TL） formulation. The numerical tests indicate that the present element is accurate and insensitive to mesh distortion.

Size-Dependent Geometrically Nonlinear Free Vibration of First-Order Shear Deformable Piezoelectric-Piezomagnetic Nanobeams Using the Nonlocal Theory

This article investigates the geometrically nonlinear free vibration of piezoelectric-piezomagnetic nanobeams subjected to magneto-electro-thermal loading taking into account size effect using the nonlocal elasticity theory.To this end,the sizedependent nonlinear governing equations of motion and corresponding boundary conditions are derived according to the nonlocal elasticity theory and the first-order shear deformation theory with von K´arm´an-type of kinematic nonlinearity.The effects of size-dependence,shear deformations,rotary inertia,piezoelectric-piezomagnetic coupling,thermal environment and geometrical nonlinearity are taken into account.The generalized differential quadrature(GDQ)method in conjunction with the numerical Galerkin method,periodic time differential operators and pseudo arclength continuation method is utilized to compute the nonlinear frequency response of piezoelectric-piezomagnetic nanobeams.The influences of various parameters such as non-dimensional nonlocal parameter,temperature change,initial applied electric voltage,initial applied magnetic potential,length-to-thickness ratio and different boundary conditions on the geometrically nonlinear free vibration characteristics of piezoelectric-piezomagnetic nanobeams are demonstrated by numerical examples.It is illustrated that the hardening spring effect increases with increasing the non-dimensional nonlocal parameter,positive initial applied voltage,negative initial applied magnetic potential,temperature rise and decreases with increasing the negative initial applied voltage,positive initial applied magnetic potential and length-tothickness ratio.

Geometrically Nonlinear Topology Optimization of Continuum Structures Based on an Independent Continuous Mapping Method

A geometrically nonlinear topology optimization method for continuum structures is proposed based on the independent continuous mapping method.The stress constraint problem is studied due to the importance of structural strength in engineering applications.First,a topology optimization model is established for a lightweight structure with element stress as constraints.Second,the stress globalization method is adopted to convert local stress constraints into strain energy constraints,which overcomes the difficulties caused by local stress constraints,such as model establishment,sensitivity analysis,and massive solution calculations.Third,the sensitivity of the objective function and constraint function is analyzed,and the method of moving asymptotes is employed to solve the optimization model.In addition,the additive hyperelasticity technique is utilized to solve the numerical instability induced by structures undergoing large deformation.Numerical examples are given to validate the feasibility of the proposed method.The method provides a significant reference for geometrically nonlinear optimization design.

基于实际预压载作业的自升式平台环境载荷图谱绘制新思路

The environmental load chart is an important technical support required for the jack-up drilling platform to facilitate its adaptation to different operating waters and ensure the safety of operation.This chart is a crucial part of the platform operation manual.The chart data are closely related to external factors such as water depth,wind,wave,and current conditions of the working water,as well as to the structural characteristics of the platform itself and the number of variable loads.This study examines the platform state under extreme wind,wave,and current conditions during preloading.In addition,this study focuses on the difference between the ultimate reaction force of the pile leg during preloading and the reaction force of the pile leg without considering any environmental load before preloading.Furthermore,the relationship between the difference and the new reaction force of the pile leg caused by the combination of different environmental conditions is established to facilitate the construction of a new form of environmental load chart.The newly formed chart is flexible and simple;thus,it can be used to evaluate the environmental adaptability of the platform in the target well location and provides the preloading target demand or variable load limit according to the given environmental constraints.Moreover,the platform can perform personalized preloading operations,thereby improving its capability to cope with complex geological conditions,such as reducing punch-through risks.This condition reduces the load on jacking system devices and increases its service life.

A SIMPLIFIED CALCULATING METHOD OF NONLINEAR FREQUENCY OF CABLE NET UNDER MEAN WIND LOAD

The cable net supported glass curtain wallas the most advanced technique in dot point supported glass curtain wall, is widely used in China. Because of its large deflection and high nonlinearity under wind load, the dynamic performance of the cable net is greatly different from that of the conventional linear structures. The continuous membrane theory is used to construct the nonlinear vibration differential equation of the cable net, and the harmonic balance method is used to solve the analytic formula of the nonlinear frequency. In order to verify the accuracy of the above analytic formula, the results of the formula and the nonlinear FEM time-history method are compared and found to be in good agreement. Furthermore, the nonlinear vibration differential equation and the nonlinear frequency obtained in this paper are the basis for the wind-induced response analysis of a cable net under fluctuating wind load.

Static and Dynamic Analysis of Mooring Lines by Nonlinear Finite Element Method

This study has focused on developing numerical procedures for the static and dynamic nonlinear analysis of mooring lines. A geometrically nonlinear finite element method using isoparametric cable element with two nodes is briefly presented on the basis of the total Lagrangian formulation. The static and dynamic equilibrium equations of mooring lines are established. An incremental-iterative method is used to determine the initial static equilibrium state of cable systems under the action of self weights, buoyancy and current. Also the Newmark method is used for dynamic nonlinear analysis of ocean cables. Numerical examples are presented to validate the present numerical method, and examine the effect of various parameters.

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.