Nodal Displacement Vector Research Articles

Evaluation of the Spatial Variability of the Mechanical Properties of Rocks Using Non‐Iterative Green's Function Approach and the FOSM Method

ABSTRACTThe present work proposes a new version of the Green‐FOSM (first‐order second moment) method, which eliminates the iterative calculation process of the original version and, simultaneously, solves the convergence problems related to the mechanical properties of rocks that form the geological formation. In this calculation scheme, the iterative process is eliminated by using a matrix that correlates the nodal displacement vector with the strain vector. Considering the same computational resources, this non‐iterative version of the Green‐FOSM method is up to 200 times faster than the original iterative process. In addition, it allows analyzing problems with more than 10,000 random variables, value that in the original method is less than 3000. To demonstrate its validity, the proposed method is applied to two hypothetical models subjected to different fluid extraction processes. For all the different levels of correlation and spatial variability, the statistical results obtained by the proposed method agree well with the results obtained via Monte Carlo Simulation (MCS). The relationship between CPU times demonstrates that the proposed method is at least 50 times faster than MCS. In the end, the non‐iterative Green‐FOSM method is used to obtain the displacement, strain, and stress fields of a geological section constructed from a seismic image of Brazilian pre‐salt oil region. The results found show that, depending on the levels of spatial variability, the analyzed fields can assume values up to 30.6% higher or lower than the values obtained deterministically.

Enhanced Strain Field Reconstruction in Ship Stiffened Panels Using Optical Fiber Sensors and the Strain Function-Inverse Finite Element Method

Accurately reconstructing the strain field within stiffened ship panels is crucial for effective structural health monitoring. This study presents a groundbreaking approach to strain field reconstruction in such panels, utilizing optical fiber sensors in conjunction with the strain function-inverse finite element method (SF-iFEM). A novel technique for solving nodal strain vectors, based on the element strain function, has been devised to improve the accuracy of strain reconstruction using the inverse finite element method (iFEM), addressing the limitations associated with traditional nodal displacement vector solutions. Moreover, the proposed method for determining the equivalent neutral layer of stiffened ship panels not only reduces the number of elements effectively but also establishes a strain function between the inner and outer surfaces of the structure. Using this function, a layout scheme for optical fiber sensors on the inner side of ship stiffened panels is provided, overcoming the symmetrical arrangement constraints of iFEM for sensor placement on both the inner and outer sides of the structure. The results demonstrate a significant improvement in strain reconstruction accuracy under bending and bending–torsion deformations compared to conventional iFEM. Consequently, the findings of this research will contribute to enhancing the engineering applicability of iFEM in ship structure health monitoring.

Smooth Lagrangian Crack Band Model Based on Spress-Sprain Relation and Lagrange Multiplier Constraint of Displacement Gradient

Abstract A preceding 2023 study argued that the resistance of a heterogeneous material to the curvature of the displacement field is the most physically realistic localization limiter for softening damage. The curvature was characterized by the second gradient of the displacement vector field, which includes the material rotation gradient, and was named the “sprain” tensor, while the term “spress” is here proposed as the force variable work-conjugate to “sprain.” The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting self-equilibrated nodal sprain forces. However, the fact that the sprain forces had to be applied on the adjacent nodes of a finite element greatly complicated the programming and extended the simulation time in a commercial code such as abaqus by almost two orders of magnitude. In the present model, Smooth Lagrangian Crack Band Model (slCBM), these computational obstacles are here overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. A crucial feature is that the nodal values of the approximate gradient tensor are shared by adjacent finite elements. The actual displacement gradient tensor calculated from the nodal displacement vectors is constrained to the approximate displacement gradient tensor by means of a Lagrange multiplier tensor, either one for each element or one for each node. The gradient tensor of the approximate gradient tensor then represents the approximate third-order displacement curvature tensor, or Hessian of the displacement field. Importantly, the Lagrange multiplier behaves as an externally applied generalized moment density that, similar to gravity, does not affect the total strain-plus-sprain energy density of material. The Helmholtz free energy of the finite element and its associated stiffness matrix are formulated and implemented in a user’s element of abaqus. The conditions of stationary values of the total free energy of the structure with respect to the nodal degrees-of-freedom yield the set of equilibrium equations of the structure for each loading step. One- and two-dimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the three-point bend test are independent of the orientation of a regular square mesh, capture the width variation of the crack band, the damage strain profile across the band, and converge as the finite element mesh is refined.

A Unit-Load Approach for Reliability-Based Design Optimization of Linear Structures under Random Loads and Boundary Conditions

The low order Taylor’s series expansion was employed in this study to estimate the reliability indices of the failure criteria for reliability-based design optimization of a linear static structure subjected to random loads and boundary conditions. By taking the advantage of the linear superposition principle, only a few analyses of the structure subjected to unit-loads are needed through the entire optimization process to produce acceptable results. Two structural examples are presented in this study to illustrate the effectiveness of the proposed approach for reliability-based design optimization: one deals with a truss structure subjected to random multiple point constraints, and the other conducts shape design optimization of a plane stress problem subjected to random point loads. Both examples were formulated and solved by the finite element method. The first example used the penalty method to reformulate the multiple point constraints as external loads, while the second example introduced an approach to propagate the uncertainty linearly from the nodal displacement vector to the nodal von Mises stress vector. The final designs obtained from the reliability-based design optimization were validated through Monte Carlo simulation. This validation process was completed with only four unit-load analyses for the first example and two for the second example.

Quadrilateral element in mixed FEM for analysis of thin shells of revolution

The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.

Volumetric element with vector approximation of the desired values for nonlinear calculation of the shell of rotation

The usage of traditional approximating functions directly to the desired displacement vector of the internal point of a finite element to determine it through nodal unknowns in the form of displacement vectors and their derivatives is described. To analyze the stress state of a geometrically non-linearly deformable shell of rotation at the loading step, the developed algorithm for forming the stiffness matrix of a hexagonal finite element with nodal values in the form of displacement increments and their derivatives was used. To obtain the desired approximating expressions, the traditional interpolation theory is used, which, when calculated in a curved coordinate system, is applied to the displacement vector of the internal point of a finite element for its approximation of class C(1) through nodal displacement vectors and their derivatives. For the coordinate transformation, expressions of the bases of nodal points are obtained in terms of the basis vectors of the inner point of the finite element. After the coordinate transformations, approximating expressions of class C(1) are found for the components of the displacement vector of the internal point of the finite element, leading in a curved coordinate system to implicitly account for the displacement of the finite element as a rigid whole. Using calculation examples, the results of the developed method of approximation of the required values of the FEM with significant displacements of the structure as an absolute solid are obtained.

Modeling a Dynamic Response at Resonant Vibrations of an Elongated Plate with an Integral Damping Coating

Classical methods of surface damping using free and constraining damping layers are discussed. The structure of a perspective integrated version of a damping coating is presented. This integral damping coating consists of two layers of a material with pronounced viscoelastic properties, between which there is a thin reinforcing layer of a high modulus material. A generalization of the Thompson-Kelvin-Voigt model is given for the description of viscoelastic properties of the material under tension-compression in the case of a complex stress state. A finite-element method was developed to determine the dynamic response of an elongated plate with the integral damping coating. This method is based on a four-layer finite element with 14 degrees of freedom: the main material is within the Kirchhoff-Love's model, the damping layers are in a flat stress state, the reinforcing layer perceives tension and compression. This model allows us to take into account the effect of transverse compression of the damping layers of the plate, which significantly increases its damping properties at high vibration frequencies. The stiffness matrices, the damping matrices, and the mass matrices of the constituent layers aim at obtaining similar complete matrices of a finite element. A system of resolving equations was obtained on the basis of the Lagrange equations of the second kind with respect to the vector of nodal displacements of the finite element model of the plate with an arbitrary dynamic load. In the case of a harmonic load with a frequency that coincides with one of the frequencies of free vibrations of the plate, a transition to a modal equation with respect to the normal coordinate corresponding to the given frequency is possible. Numerical experiments were carried out to test the developed finite element method using the example of a hingedly supported elongated plate with an integral damping coating. The numerical experiments showed a qualitative change in the composition of stresses in the damping layers of the plate at high vibration frequencies, which significantly affects its damping properties.

An effective topology optimization method for crashworthiness of thin-walled structures using the equivalent linear static loads

The equivalent static loads method for nonlinear dynamic response structural optimization may be failed in large deformation crash conditions, due to topology optimization with the equivalent static loads mostly beyond the linear range and causing numerical defects such as high compliance of elements. To overcome the above disadvantage, an advanced structural topology optimization method for crashworthiness considering crash-reduced large deformation and plastic buckling is proposed using newly defined equivalent linear static loads. The equivalent linear static loads can adaptively scale to guarantee that the topology optimization is performed within linear range. At each cycle, the crash simulation is performed and the nonlinear nodal displacement vector at the time step with the maximum strain energy is scaled by an adaptive displacement-scaling factor. The equivalent linear static loads that are generated by multiplying the linear stiffness matrix and the scaled nodal displacement vector will be incorporated into topology optimization, which can guarantee the topology optimization to remain in linear range and further solve the numerical instability problems. The process is repeated until the convergence criteria are satisfied. The effectiveness of the proposed method is evaluated by solving a crashworthiness topology optimization of a crash box considering crash-induced plastic buckling to determine the location and profile of crash triggers. The results show that the proposed method can effectively solve the large deformation crashworthiness topology optimization of thin-walled structures and provides a feasible strategy for crash triggers design in crash box.

Full-Space Response Surface Method for Analysis of Structural Reliability

In order to improve the computational efficiency of the response surface method (RSM) in analysis of structural reliability, a full-space response surface method (FRSM) is presented in this paper. First, a vector-type response surface is developed by expanding the stochastic nodal displacement vector along the Krylov basis vectors defined by the global stiffness matrix and the force vector. Then the effective collocation points are picked out of the candidate ones according to the linear independence of the combining row vector. Finally, the unknown coefficients of the proposed response surface are determined by means of the regression analysis. Examples show that the proposed FRSM requires much fewer effective collocation points and less times of finite element analysis, achieving much higher computational efficiency by comparing with the traditional RSM.

Inferring microseismic source mechanisms and in situ stresses during triaxial deformation of a North-Sea-analogue sandstone

Abstract. Monitoring microseismic activity provides a window through which to observe reservoir deformation during hydrocarbon and geothermal energy production, or CO2 injection and storage. Specifically, microseismic monitoring may help constrain geomechanical models through an improved understanding of the location and geometry of faults, and the stress conditions local to them. Such techniques can be assessed in the laboratory, where fault geometries and stress conditions are well constrained. We carried out a triaxial test on a sample of Red Wildmoor sandstone, an analogue to a weak North Sea reservoir sandstone. The sample was coupled with an array of piezo-transducers, to measure ultrasonic wave velocities and monitor acoustic emissions (AE) – sample-scale microseismic activity associated with micro-cracking. We calculated the rate of AE, localised the AE events, and inferred their moment tensor from P-wave first motion polarities and amplitudes. We applied a biaxial decomposition to the resulting moment tensors of the high signal-to-noise ratio events, to provide nodal planes, slip vectors, and displacement vectors for each event. These attributes were then used to infer local stress directions and their relative magnitudes. Both the AE fracture mechanisms and the inferred stress conditions correspond to the sample-scale fracturing and applied stresses. This workflow, which considers fracture models relevant to the subsurface, can be applied to large-scale geoengineering applications to obtain fracture mechanisms and in-situ stresses from recorded microseismic data.

Geometric nonlinear formulation of plate buckling structure

The analytical or exact mathematical formulation of stresses and displacements for plate buckling structure become impossible to develop if the plate geometry is so complicated. Numerical technique is one another approach to solve this problem and it is chosen in this study for plate structure under in-plane and out-plane load. The formulation of elastic stiffness matrix (ke) and geometric nonlinear stiffness matrix (kg) of the plate structure due to buckling is presented and based on virtual displacement principle. The geometric nonlinear stiffness matrix (kg) is found function of internal stresses. The direct iteration technique is applied to find nodal displacements. Under this technique, the Gauss points stresses are initialized as zeros, then the kg matrix is updated, and then a new nodal displacement vector is found for the next approximation of internal stresses. Iterative process is done until convergence of displacement is satisfied. The rectangular plate with one fixed edge supported is used to test the proposed nonlinear formulation and procedure. The compressive in-plane load and moment is considered and applied for the tested plate. The plate is discretized with appropriate number of triangular finite element mesh. It is found that, the convergence of displacement is satisfied by using direct iteration technique. The load - deflection curve shows nonlinear relationship and approach to critical load. This finding shows that the direct iteration method can be accepted for the analyzing of plate buckling by considering geometric nonlinear assumption.

Of the matrics of the equilibrium of the rod formation system based on the principle the duality of the problems of structures mechanics

The formation of the equilibrium matrix of the core system in the matrix form is based on the use of the mechanical model of the system obtained by its discretization. The topological structure of the model is set using the graph and the accompanying incidence matrix. The matrix transformation of the vector of nodal displacements in combination with the extended incidence matrix allows determining the absolute elongations and distortions of each finite element. The composition of only two matrices (matrix of incidence and lengths of elements) and the skew vector leads to a geometric matrix characterizing the dependence of concentrated bending deformations in the calculated cross sections of the core system from the nodal displacements for a given load. Based on the duality principle, by transposing the geometric matrix, the equilibrium equation of the core system is derived in matrix form.

Linear Analysis of Stability of Pitched Roof Frames

Background: In this paper, geometric nonlinear analysis of pitched roof frames was carried out by the stiffness matrix method using stability functions. Objective: This study contributes to a better knowledge of the stability of pitched roof frames, not braced, and therefore of the efficiency in their dimensioning. Method: At first, the argument of the stability functions was set as 0.01. The stiffness matrix of the frame has been assembled, as well as the nodal load vector of the frame. The boundary conditions (support restraint and wind bracing restraint) were introduced for the reduction of this matrix and the nodal load vector. At this stage, the determinant of the reduced stiffness matrix and the reduced nodal displacement vector are calculated. The argument of the stability functions is incremented by 0.01 and the operations are repeated until the determinant of the reduced stiffness matrix changes sign. The argument of the iteration preceding the sign change of the determinant and corresponding to its positive value is taken and refined by a process described in the paper. The buckling loads of the frame members are determined at this stage. Results and Conclusion: The analysis focused on four frames; the obtained results show that the increase in the inclination of the crossbar makes it possible to take full advantage of the “arch effect”. Arch effect is due to the presence of crossbars which have a linear arch shape. Furthermore, the angle as well as the length ratio, between the crossbar and post, influence critical load value.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenvalue problems. Finally, we investigate the convergence of Lagrange and B-spline approaches when the number of degrees of freedom grows. The study shows that cubic B-spline functions provide the best solution with the smallest relative errors for a given number of degrees of freedom.

Improved step-by-step modeling method for analyzing the mechanical behavior of steel structures during construction

The step-by-step modeling method considering nonlinear effects is an effective method for analyzing the mechanical behavior of steel structures during construction. However, two problems have limited the widespread application of the traditional step-by-step modeling method: the positioning of newly assembled members and the transformation of the stiffness matrix and the nodal load and displacement vectors between different stages of construction. In this article, a new concept of “two moments of a construction stage” (the initial and final moments of each construction stage) is proposed to improve the step-by-step modeling method. Based on this concept, an improved step-by-step modeling method, which considers the modified design configuration positioning principle for newly assembled members and provides a method for modifying the structural stiffness matrix at the initial moment of any construction stage, is proposed. A calculation program block based on the proposed method is compiled to analyze the mechanical behavior of a plane frame and a plane shallow arch during construction. The mechanical behavior of an engineering application, the connective corridor of Shanghai International Financial Centre, is also analyzed using the proposed method, and the calculated results are compared with the monitoring results. The numerical results show that the modified design configuration positioning principle is applicable for the positioning of newly assembled members and that the modification method for the structural stiffness matrix and the nodal load and displacement vectors at the initial moment of a construction stage efficiently solves the problem of their transformation between different stages of construction. The improved step-by-step modeling method proposed in this article is more valid and accurate than other methods for analyzing the mechanical behavior of steel structures during construction.

Analytical Stiffness Matrix Method for Flexural Vibration of Compression Bar

According to differential equation for transverse flexural vibration of compression bar considering second-order effect and inertia force, displacement vector expression was achieved. Based on the displacement boundary condition, displacement coefficient expressed by nodal displacement vector was obtained. Internal force equations of compression bar were established and then internal force at bar ends expressed by nodal displacement vector was provided. Finally, dynamic stiffness matrix colligating mass matrix and geometry matrix was given and can be applied to accurate analysis for dynamic performance and dynamic responses of bar.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenval...

Two‐scale topology optimization for composite plates with in‐plane periodicity

SummaryThis study proposes a two‐scale topology optimization method for a microstructure (an in‐plane unit cell) that maximizes the macroscopic mechanical performance of composite plates. The proposed method is based on the in‐plane homogenization method for a composite plate model in which the macrostructure is modeled using thick plate theory and the microstructures are three‐dimensional solids. Macroscopic plate characteristics such as homogenized plate stiffnesses and generalized thermal strains are evaluated through the application of numerical plate tests applied to an in‐plane unit cell. To handle large rotations of the composite plates, we employ a co‐rotational formulation that facilitates working with the two‐scale plate model formulated within a small strain framework. Two types of objective functions are tested in the presented optimization problems: one minimizes the macroscopic end compliance to maximize the macroscopic plate stiffness, whereas the other maximizes components of a macroscopic nodal displacement vector. Analytical sensitivities are derived based on in‐plane homogenization formulae so that a gradient‐based method can be employed to update the topology of in‐plane unit cells. Several numerical examples are presented to demonstrate the proposed method's capability related to the design of optimal in‐plane unit cells of composite plates. Copyright © 2017 John Wiley & Sons, Ltd.

Numerical algorithm for solving of nonlinear problems of structural mechanics based on the continuation method in combination with the dynamic relaxation method

A numerical algorithm of strength and stability analysis of nonlinear deformable bar systems and thin-walled spatial structures is proposed. A numerical technique is based on the continuation method and the dynamic relaxation method. When using the method of dynamic relaxation the state of static equilibrium of structures is defined after the damped oscillations by integrating over leading parameter. The solution of the initial equation system describing the motion of the mechanical system is reduced to the solution of Cauchy problem for systems of ordinary differential equations. At an each step in the leading parameter the vector of nodal displacements and the time parameter are defined. Several examples of numerical analysis for bar, shell and plate are given.

Physical Constraint Finite Element Model for Medical Image Registration.

Due to being derived from linear assumption, most elastic body based non-rigid image registration algorithms are facing challenges for soft tissues with complex nonlinear behavior and with large deformations. To take into account the geometric nonlinearity of soft tissues, we propose a registration algorithm on the basis of Newtonian differential equation. The material behavior of soft tissues is modeled as St. Venant-Kirchhoff elasticity, and the nonlinearity of the continuum represents the quadratic term of the deformation gradient under the Green- St.Venant strain. In our algorithm, the elastic force is formulated as the derivative of the deformation energy with respect to the nodal displacement vectors of the finite element; the external force is determined by the registration similarity gradient flow which drives the floating image deforming to the equilibrium condition. We compared our approach to three other models: 1) the conventional linear elastic finite element model (FEM); 2) the dynamic elastic FEM; 3) the robust block matching (RBM) method. The registration accuracy was measured using three similarities: MSD (Mean Square Difference), NC (Normalized Correlation) and NMI (Normalized Mutual Information), and was also measured using the mean and max distance between the ground seeds and corresponding ones after registration. We validated our method on 60 image pairs including 30 medical image pairs with artificial deformation and 30 clinical image pairs for both the chest chemotherapy treatment in different periods and brain MRI normalization. Our method achieved a distance error of 0.320±0.138 mm in x direction and 0.326±0.111 mm in y direction, MSD of 41.96±13.74, NC of 0.9958±0.0019, NMI of 1.2962±0.0114 for images with large artificial deformations; and average NC of 0.9622±0.008 and NMI of 1.2764±0.0089 for the real clinical cases. Student’s t-test demonstrated that our model statistically outperformed the other methods in comparison (p-values <0.05).