Shape Functions Research Articles

The Thermodynamic Responses of FG Conical/Cylindrical/Annular Panels with Circumferentially Variable Dimensions

This paper is the first to develop a meshless thermodynamic model with circumferentially variable dimensions to analyze the thermal-vibration mechanisms of typical functionally graded (FG) panel-type structures, mainly including conical, cylindrical, and annular panels. The longitudinal variation of the panel-type structure’s circumferential dimensions is taken into account according to specific rules. The Thin Plate Inverse Mapping (TPIM) meshless shape functions are introduced to characterize the displacement components associated with each substructure. Thermodynamic equations are formulated based on FSDT thermo-elastic theory as well as Hamilton’s principle. Rectangular and exponential pulse loads are taken into consideration for forced vibration analysis in this formulation. Numerical convergence and comparative cases are provided. Solutions from the literature and finite element simulation results are utilized for comparison, revealing that the percentage differences observed are all less than 0.025%. This confirms the reliability and accuracy of the developed analysis model. In conclusion, a detailed investigation is conducted on the influences of geometrical dimensions, boundary conditions, as well as thermal and external loads on the thermodynamic behaviors of FG panel-type structures with circumferentially variable dimensions.

Prestress Force and General Excitation Identification for Plate-like Concrete Bridges

Prestress force dominates the carrying capacity of concrete girders, and is vital for bridge health monitoring. Many vibrational-based methods have been proposed to determine prestress using bridge responses induced by known excitations, which means that they can barely use the normal traffic of in-service bridges as excitation to achieve long-term monitoring. Moreover, most studies are based on beam theory, which may not be precise for plate-like bridges. Hence, this paper establishes a motion equation for a prestressed slab via Kirchhoff’s plate theory and proposes a two-step procedure to assess the prestress and general excitation simultaneously through only bridge responses. The excitation is determined in the first step via the Load Shape function method and used as input for the prestress identification via state-space formulation in the second step. A numerical study on a prestressed plate subjected to a moving load is conducted. Considering different levels of measurement noise and load speed, the proposed method can determine the prestress and moving load with 7.33% and 10.18% error, respectively. A laboratory test on a prestressed box girder subjected to a fixed cyclic load is performed, the prestress and cyclic load are both determined to have good stability, and the errors are under 11.32%.

A projection‐based quaternion discretization of the geometrically exact beam model

AbstractIn the present work the geometrically exact beam model is formulated in terms of unit quaternions. A projection‐based discretization approach is proposed which is based on a normalization of the quaternion approximation. The discretization relies on NURBS shape functions and, alternatively, on Lagrangian interpolation. The redundancy of the quaternions is resolved by applying the method of Lagrange multipliers. In a second step the Lagrange multipliers are eliminated circumventing the need to solve saddle point systems. The resulting finite elements retain the objectivity of the underlying beam formulation. Optimal rates of convergence are observed in representative numerical examples.

Tracking longitudinal thalamic volume changes during early stages of SCA1 and SCA2

PurposeSpinocerebellar ataxia SCA1 and SCA2 are adult-onset hereditary disorders, due to triplet CAG expansion in their respective causative genes. The pathophysiology of SCA1 and SCA2 suggests alterations of cerebello-thalamo-cortical pathway and its connections to the basal ganglia. In this framework, thalamic integrity is crucial for shaping efficient whole-brain dynamics and functions. The aims of the study are to identify structural changes in thalamic nuclei in presymptomatic and symptomatic SCA1 and SCA2 patients and to assess disease progression within a 1-year interval.Material and methodsA prospective 1-year clinical and MRI assessment was conducted in 27 presymptomatic and 23 clinically manifest mutation carriers for SCA1 and SCA2 expansions. Cross-sectional and longitudinal changes of thalamic nuclei volume were investigated in SCA1 and SCA2 individuals and in healthy participants (n = 20).ResultsBoth SCA1 and SCA2 patients had significant atrophy in the majority of thalamic nuclei, except for the posterior and partly medial nuclei. The 1-year longitudinal evaluation showed a specific pattern of atrophy in ventral and posterior thalamus, detectable even at the presymptomatic stage of the disease.ConclusionFor the first time in vivo, our exploratory study has shown that different thalamic nuclei are involved at different stages of the degenerative process in both SCA1 and SCA2. It is therefore possible that thalamic alterations might significantly contribute to the progression of the disease years before overt clinical manifestations occur.

Possibility of the Traversable Wormholes in the Galactic Halos within 4D Einstein–Gauss–Bonnet Gravity

AbstractRecently, there has been significant interest regarding the regularization of a limit of Einstein–Gauss–Bonnet (EGB) gravity. This regularization involves re‐scaling the Gauss–Bonnet (GB) coupling constant as , which bypasses Lovelock's theorem and avoids Ostrogradsky instability. A noteworthy observation is that the maximally or spherically symmetric solutions for all the regularized gravities coincide in the scenario. Considering this, the wormhole solutions in the galactic halos are investigated based on three different choices of dark matter (DM) profiles, such as Universal Rotation Curve, Navarro–Frenk–White, and Scalar Field Dark Matter with the framework of EGB gravity. Also, the Karmarkar condition is used to find the exact solutions for the shape functions under different non‐constant redshift functions. The energy conditions for each DM profile are discussed and the influence of GB coefficient in violating energy conditions are noticed, especially null energy conditions. Further, some physical features of wormholes, viz. complexity factor, active gravitational mass, total gravitational energy, and embedding diagrams, have been explored.

Guidelines for element size and type selection for the finite element simulation of laser-induced elastic waves in thermoelastic laser ultrasonic testing

This paper explores spatial discretization within finite element simulations of laser-induced elastic waves within the context of Laser Ultrasonic Testing (LUT). Motivated by discrepancies and oscillations detected in temperature and displacement results in the literature, we traced these issues back to spatial discretization challenges. These challenges originate from rapid localized heating and the generation and propagation of high-frequency waves across a relatively large domain. This study effectively addresses and rectifies these inaccuracies, offering guidance for selecting the appropriate element size and type. We examined two element types: four-node quadrilaterals (Q4) employing first-order Lagrange and nine-node quadrilaterals (Q9) using second-order Lagrange shape functions. Our analysis encompasses mesh refinement strategies, exploration of time and frequency domain plots for temperature and displacement, as well as an evaluation of different pulse durations. Our findings demonstrate that Q9 elements attain accuracy with grids four times larger than Q4 elements for temperature and wave propagation analyses. Furthermore, we observe that lower frequency waves exhibit reduced sensitivity to element size, emphasizing the relationship between element size and elastic wave frequency. Pulse durations in the 6 to 30 ns range affect the required element size in the heat-affected zone but exert minimal influence on wave frequency and spatial discretization in the remainder of the domain. Finally, we present a new formula for element size selection based on the dominant frequency. This study provides a comprehensive guideline for selecting element size and type, enabling the attainment of accurate results while effectively managing computational costs.

Conformal wormholes in f(R,Lm) theory

In this study, we present a new static and conformally symmetric wormhole solution. We develop new wormhole model in f(R,Lm) gravity. Shape function of conformally symmetric f(R,Lm) wormholes, provided by homothetic vector field is obtained regarding to conformal factor. It is investigated whether the function provides a stable throat infrastructure that does not allow collapsing. Anisotropic fluid matter filled the obtained conformally symmetric traversable wormholes is discussed by examining the dynamical properties of the fluid. Constructed new wormhole model violates all energy conditions. Physical significance of the sources filled constructed wormholes is discussed using volume quantifier. It is revealed that f(R,Lm) theory allows traversable and conformally symmetric wormholes if supported by phantom field.

Free vibration analysis of nanocomposite plate in contact with fluid using a novel quasi-3D shear deformation theory and artificial neural network

This article presents an analytical approach leveraging a novel quasi-three-dimensional shear deformation theory to analyze the free vibration response of functionally graded graphene platelet-reinforced composite (FG-GPLRC) plates in a liquid medium. The study meticulously determines material properties using the Halpin-Tsai micromechanical model, offering a thorough characterization of these advanced nanocomposite structures. Differential motion equations, incorporating transverse shear and normal stresses, are derived via Hamilton’s principle. The natural frequencies of the nanocomposite plates are computed using Galerkin’s method and validated against published results, affirming their accuracy and reliability. The key innovation of this research is introducing a new trigonometric shape function, which exhibits superior precision compared to previously proposed shape functions and existing theories. Furthermore, an artificial neural network (ANN) model is developed to train the computed results, enabling precise prediction of the natural frequencies without further computational runs. The Bayesian Regularization algorithm, implemented in Matlab, is utilized for this purpose. Across different GPLs patterns and varying input parameters, the error percentage between the ANN model predictions and the results from the Galerkin’s method is consistently low, often below 0.1%. The ANN model demonstrates robust generalization by effectively predicting outcomes with input data beyond its training range. Additionally, it provides immediate computational results, eliminating the delays of traditional methods. The article also examines the effects of various factors, including material characteristics, geometric parameters, and the fluid medium, on the free vibration behavior of FG-GPLRC plates.

Free vibration analysis of rod systems using the finite element force method with extended shape functions

For free vibration analysis of rod systems formulation of the finite element force method with extended shape functions obtained by using the mode shapes of a fixed-fixed rod is proposed. Examples illustrating the accuracy of numerical solutions for three types of finite elements are given. The use of proposed approach for the space frame gave differences between numerical solutions dealt with 8 finite elements and exact results less than 5% for the first five natural frequencies. Для анализа свободных колебаний стержневых систем предложена формулировка метода конечных элементов в усилиях с расширенными функциями формы, полученными с использованием форм колебаний неподвижно закрепленного стержня. Приведены примеры, иллюстрирующие точность численного решения для трех типов конечных элементов. Использование предложенного подхода для пространственной рамы дало различия между численными решениями для восьми конечных элементов и точными результатами менее 5% для первых пяти собственных частот.

Probing the existence of wormhole solutions in [formula omitted] gravity with conformal symmetry

This study presents novel and physically plausible wormhole solutions within the framework of f(Q,T) gravity theory, incorporating conformal symmetries. The investigation explores the feasibility of traversable wormholes under diverse scenarios, considering traceless, anisotropic, and barotropic equations of state. Additionally, the influence of model parameters on the existence and characteristics of these wormhole structures is thoroughly examined. Notably, the derived shape function satisfies all the necessary criteria. Furthermore, in one of the cases, the presence of non-exotic fluid is confirmed, while in others, exotic matter is only required near the wormhole throat.

Highly efficient inverse lumped modeling for the pre-strained circular dielectric elastomer

The equi-biaxially pre-strained circular dielectric elastomer actuator (PCDEA) is a classical actuator configuration to demonstrate the electromechanical deformation mechanism of dielectric elastomer. Till now, there still lacks an effective method to analyze the non-uniform voltage-induced deformation of PCDEA with both high accuracy and low computational cost. Besides, the inverse problem that determines the voltage input for a desired deformation is also a hard problem for the current modeling approaches. In this work, a novel shape function approximation method is proposed to model the non-uniform deformation of PCDEA in a highly accurate lumped-parameter way. The method can solve both the inverse problem and forward problem of the PCDEA, making it possible to position the onset of electromechanical instability (EMI) of this configuration. Detailed comparisons are made between this method and high-resolution finite element analysis to verify its accuracy. This method also features low computational cost, providing opportunities for developing model-based real-time control and proprioception techniques. Further investigation is also carried out on how material types and configurational parameters interact with PCDEAs’ performance, along with checking the findings with experiment observation. Guidelines for DE material development and actuator design are also summarized accordingly.

Variational temporal convolutional networks for I-FENN thermoelasticity

Machine learning (ML) has been used to solve multiphysics problems like thermoelasticity through multi-layer perceptron (MLP) networks. However, MLPs have high computational costs and need to be trained for each prediction instance. To overcome these limitations, we introduced an integrated finite element neural network (I-FENN) framework to solve transient thermoelasticity problems in Abueidda and Mobasher (2024). This approach used a physics-informed temporal convolutional network (PI-TCN) within a finite element scheme for solving transient thermoelasticity problems. In this paper, we introduce an I-FENN framework using a new variational TCN model trained to minimize the thermoelastic variational form rather than the strong form of the energy balance. We mathematically prove that the I-FENN setup based on minimizing the variational form of transient thermoelasticity still leads to the same solution as the strong form. Introducing the variational form to the ML model brings the advantages of lower requirement for the differentiability of the basis function and, thus, lower memory requirement and higher computational efficiency. Also, it automatically satisfies zero Neumann boundary conditions, thus reducing the complexity of the loss function. The formulation based on the variational form complies with thermodynamic requirements. The proposed loss function reduces the difference between predicted and target data while minimizing the variational form of thermoelasticity equations, combining the benefits of both data-driven and variational methods. In addition, this study uses finite element shape functions for spatial gradient calculations and compares their performance against automatic differentiation. Our results reveal that models leveraging shape functions exhibit higher accuracy in capturing the behavior of the thermoelasticity problem and faster convergence. Adding the variational term and using shape functions for gradient calculations ensure better adherence to the underlying physics. We demonstrate the capabilities of this I-FENN framework through multiple numerical examples. Additionally, we discuss the convergence of the proposed variational TCN model and the impact of hyperparameters on its performance. The proposed approach offers a well-founded and flexible platform for solving fully coupled thermoelasticity problems while retaining computational efficiency, where the efficiency scales proportional to the model size.

Exploring wormhole solutions with global monopole charge in the context of f(Q) gravity

This study explores the potential existence of traversable wormholes influenced by a global monopole charge within the f(Q) gravity framework. To elucidate the characteristics of these wormholes, we conducted a comprehensive analysis of wormhole solutions employing three different forms of redshift function under a linear f(Q) model. Wormhole shape functions were derived for barotropic, anisotropic, and isotropic Equations of State (EoS) cases. However, in the isotropic EoS case, the calculated shape function failed to satisfy the asymptotic flatness condition. Additionally, we observed that our obtained shape functions adhered to the flaring-out conditions under an asymptotic background for the remaining EoS cases. Furthermore, we examined the energy conditions at the wormhole throat with a radius r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document}. We noted the influences of the global monopole’s parameter η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}, the EoS parameter ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}, and n in violating energy conditions, particularly the null energy conditions. Finally, we conducted a stability analysis utilizing the Tolman–Oppenheimer–Volkov (TOV) equation and found that our obtained wormhole solution is stable.

Modeling the measurement accuracy of one-dimensional boundary subsets in digital image correlation

Boundary subsets, which contain unwanted pixels from background image or other regions, generally occur at areas of vital mechanical importance. In this work, we establish a theoretical model characterizing the measurement accuracy of 1-dimensional boundary subsets, which explicitly expresses the retrieved displacement in terms of subset size, invalid pixels, shape function order, and underlying deformation field. Particularly, the transfer function of digital image correlation at boundary subsets is derived and verified, indicating possible amplification of amplitude and the existence of phase shift, which are quite different from standard full square subsets. Finally, the retrieved deformation near boundary is theoretically analyzed, and closed-form formulae are deduced for polynomial underlying displacements in consideration of the Weierstrass approximation theorem. To the author's knowledge, this is the first general mathematical model for boundary subsets, and paves the way for enhanced measurement accuracy.

A semi-analytical spectral element model for guided wave propagation in composite laminated conical shells

The conical shell is a typical non-uniform waveguide with its circumferential radius varies linearly. This work aims to provide physical insights for the composite laminated conical shell in terms of waves. A semi-analytical spectral element model is presented, which adopts the Fourier series in the circumferential direction and the higher-order shape functions in the axial direction. The first-order shear deformation theory and Hamilton principle are employed to derive the energy functions of the conical shell, and they are discretized using the spectral elements to achieve a weak-form variational problem. The dispersion equation and the group velocity expression are derived via the piecewise approximation concept and Bloch's periodic theorem. The accuracy of the developed spectral element is verified by comparison with published results. The deformation evolution mechanisms of the composite conical shell are revealed, and the wave propagation characteristics of uniform and ply drop-off composite conical shells are investigated. The results demonstrated that wavenumber distribution at 0–5 kHz is spatially dependent, and wavenumber becomes cut off and bifurcate near the cone vertex. When the frequency exceeds 2 kHz, the deformation of conical shell changes from rigid to flexible, and the structure tends to be a uniform waveguide. Compared to the geometric curvature effect, the asymmetric lay-up leads to more significant mode coupling.

Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as the uniform ball property, which ensures \mathscr{C}^{1,1} regularity of the hypersurface. In this paper, we revisit and generalize the results of Dalphin (2018 and 2020) and Guo and Yang (2013). We provide a simpler framework and more concise proofs of some of the results contained in these references and extend them to a new class of problems involving PDEs. Indeed, by using the signed distance, we avoid the intensive and technical use of local maps, as was the case in the above references. Our approach, originally developed to solve an existence problem in Privat, Robin, and Sigalotti’s 2022 paper, can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.

Theoretical nonlinear force-displacement constitutive model of triangular-plate steel damper

Utilizing the inelastic deformation of a steel damper to dissipate seismic energy and control structural damage is a prevalent design strategy in a passive control field. Such a design strategy requires an explicit force-displacement constitutive model of the steel damper in both the elastic and plastic deformation phases. Currently, the nonlinear force-displacement relationship of the steel damper is mainly determined by quasi-static tests or numerical simulations which are generally time-consuming, and there is a lack of a theoretical and straightforward approach to determine the nonlinear force-displacement constitutive model of the steel damper. This study thus theoretically derives the nonlinear force-displacement constitutive model for a commonly used triangular-plate steel damper. The theoretical formula is derived based on the equal curvature assumption commonly applied to the triangular plate and is further modified based on the curvature shape function proposed in this study for the first time. After that, the force-displacement curves obtained from the theoretical formula are compared with those obtained from a large number of numerical simulations and several sets of quasi-static tests. The result shows that the theoretical formula derived based on the curvature shape function can better predict the nonlinear force-displacement curve of the triangular-plate steel damper, which significantly outperforms the theoretical formula derived based on the equal curvature assumption especially as the triangular-plate steel damper has a large plastic deformation. Furthermore, the influence factors of the strain-hardening phenomenon existing in the force-displacement curve of the triangular-plate steel damper are analyzed based on the modified theoretical formula. The result implies that the thickness-to-height ratio of the plate is the most significant factor for the strain-hardening behavior of the triangular-plate steel damper.

Hybrid meshless-FEM method for 3-D magnetotelluric modelling using non-conformal discretization

SUMMARY We propose a novel method for 3-D magnetotelluric (MT) forward modelling based on hybrid meshless and finite-element (FE) methods. This method divides the earth model into a central computational region and an expansion one. For the central region, we adopt scatter points to discretize the model, which can flexibly and accurately characterize the complex structures without generating unstructured mesh. The meshless method using compact support radial basis function is applied to simulate this area's electromagnetic field. While in the expansion region, to avoid the heavy time consumption and numerical error of the meshless method caused by non-uniform nodes, we adopt a node-based finite-element method with regular hexahedral mesh for stability. Finally, the two discretized systems are coupled at the interface nodes according to the continuity conditions of vector and scalar potentials. Considering that the normal electric field is discontinuous at the interface with resistivity discontinuity, while the shape functions for the meshless method are continuous, we further adopt the visibility criterion in constructing the support region. Numerical experiments on typical models show that using the same degree of freedom, the hybrid meshless-finite element method (FEM) algorithm has higher accuracy than the node-based FEM and meshless method. In addition, the electric field discontinuity at interfaces is well preserved, which proves the effectiveness of the visibility criterion method. In general, compared to the conventional grid-based method, this new approach does not need the complex mesh generation for complex structures and can achieve high accuracy, thus it has the potential to become a powerful 3-D MT forward modelling technique.

Wormholes in the f(R,L,T) theory of gravity

Morris and Thorne developed wormhole solutions in the late 1980s when they discovered a recipe that wormholes must follow for travelers to cross them safely. They describe exotic matter as satisfying −pr>ρ, where pr is the radial pressure and ρ is the energy density of the wormhole. This is a notable characteristic of the General Relativity Theory. The current article discusses traversable wormhole solutions in f(R,L,T)=R+αL+βT, with α and β are model parameters. The wormhole solutions presented here satisfy the metric constraints of traversability while remarkably avoiding the exotic matter condition, indicating that f(R,L,T) gravity wormholes can be filled with ordinary matter. The derived solutions for the shape function of the wormhole meet the required metric conditions. They exhibit behavior that is comparable to that of wormholes reported in earlier references, which is also the case for our solutions for the energy density of such objects.

Stability analysis of wormhole solutions in [formula omitted] gravity utilizing Karmarkar condition with radial dependent redshift function

In this work, we examine the properties of wormhole (WH) solutions in the context of modified f(Q) gravity. The shape function (ξS(r)) and redshift function (ζ(r)) are crucial in wormhole modeling since they determine the metric of a wormhole and define its attributes. The investigation provides the interaction between f(Q) gravity features and gravitational effects, guaranteeing insights into potential wormhole configurations. Subsequently, we have looked at the behavior of energy conditions and the fundamental properties of WHs. Additionally, the volume integral quantifier (V IQ) is addressed, providing insight into the quantity of exotic matter needed in the vicinity of the wormhole throat. Furthermore, the stability of the WH solutions in both cases is examined using the Tolman–Oppenheimer–Volkoff (TOV) equation where it is evident that both WH solutions meet the equilibrium requirements. Both cases ensures the real-world acceptability of WH solutions due to the rising nature of active gravitational mass (Mactive). Our findings contribute to the current discussion about alternative gravity theories and exotic spacetime geometries.