Finite Element Method Solution Research Articles

Geometrically nonlinear analysis of planar frame structures with positional finite element method

Problems in structural engineering involving geometric nonlinearity are widely studied from numerical and experimental perspectives. From the numerical modeling point of view, to investigate the resistance mechanisms that develop in structures when subjected to large displacements, a mathematical formulation capable of numerically representing the phenomena that occur during the nonlinear behavior is necessary. This work addresses problems of planar frame structures involving geometric nonlinearity with a formulation available in the literature of the Finite Element Method (FEM) based on positions and generalized vectors as degrees of freedom. In this formulation, nodal positions are used as degrees of freedom of the problem, and the Saint-Venant-Kirchhoff constitutive law for plane stress state is adopted. Unlike the FEM formulation for displacements, the positional formulation adopted requires strategies for connecting non-collinear finite elements, involving penalization techniques to satisfy boundary conditions. As detailed in the literature the connections between non-collinear elements are performed with uniaxial and flexural springs. The strain energies of the springs are calculated based on the nodal positions of the structure, and their contributions to the Hessian matrix and internal force vector of the structure are performed with a penalization technique. To validate the implementation developed the solutions of the positional FEM are compared to analytical and numerical solutions obtained by the finite element software DIANA FEA. In addition, parametric analyses varying the connections stiffness values are carried out to determine minimum stiffness values for the connection springs to allow the representation of the nonlinear equilibrium trajectory of planar frame structures.

Boundary element method for buckling analysis of elastically connected double-beam system

Studies of Double-beam systems have received significant attention from researchers due to their wide applications in civil and mechanical engineering. Commonly, finite element method or exact solutions have been employed to solve double-beam systems, with few others considering buckling of axially loaded double-beam systems with classical boundary conditions. This paper presents a novel formulation of the Boundary Element Method (BEM) to determine the critical buckling load of the double-beam system elastically connected by a Winkler elastic layer with generalized boundary conditions. Based on the Euler-Bernoulli beam theory, the double-beam system is composed of two identical beams. This paper provides a detailed discussion of each step involved in the BEM, including the fundamental solution, boundary integral equations, and algebraic system. Examples considering different boundary conditions, material properties, and load cases are done and compared to corresponding analytical solution. The results show excellent agreement between the BEM approach and the analytical solution, confirming the accuracy and effectiveness of the technique.

Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs

Abstract In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

Identifying the mixed and forced convection flow with transverse magnetic field

AbstractOptimal control of the steady, laminar, magnetohydrodynamics (MHD) mixed convection flow of an electrically conducting fluid is considered under the effect of the transverse magnetic field in a square duct. The viscous and Joule dissipations are included and the flow is driven by a constant pressure gradient. The triple nonlinear set of momentum, induction, and energy equations are solved in dimensionless form by using the mixed finite element method (FEM) with the implementation of Newton's method for nonlinearity with the discretize‐then‐linearize approach. Accordingly, FEM solutions are obtained for various values of the problem parameters to ensure the efficiency of the underlying scheme. This study aims to investigate the problem of controlling the steady flow by using the physically significant parameters of the problem as control variables in the case of a mixed convection flow. In this respect, classification of the type of convection, forced or free, is achieved by controlling the Grashof number (Gr). Besides, single and pairwise controls with Hartmann number (M), Prandtl number (Pr), and magnetic Reynolds number (Rm) are also used to regain the prescribed fluid behaviors and required magnetic field. Control simulations are conducted with the sequential‐least‐squares‐programming (SLSQP) algorithm in the optimization.

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

In this paper, we discuss a von Mises plasticity model with nonlinear isotropic hardening assuming small strains in a plane strain example of internally pressurised thick-walled cylinder subjected to different loading conditions. The elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress exceeds the yield stress, corrections are made locally through a return mapping algorithm. We present a novel method that uses a Radial Basis Function-Finite Difference (RBF-FD) approach with Picard iteration to solve the system of nonlinear equations arising from plastic deformation. This technique eliminates the need to stabilise the divergence operator and avoids special positioning of the boundary nodes, while preserving the elegance of the meshless discretisation and avoiding the introduction of new parameters that would require tuning. The results of the proposed method are compared with analytical and Finite Element Method (FEM) solutions. The results show that the proposed method achieves comparable accuracy to FEM while offering significant advantages in the treatment of complex geometries without the need for conventional meshing or special treatment of boundary nodes or differential operators.

Mixed finite element method for multi-layer elastic contact systems

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is provided. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter H is verified.

Analysis on transient wave propagation in the soft matter of hydrogels

The precise control and structural design of the soft matter of the hydrogel subjected to dynamic loading require an in-depth understanding of its transient wave characteristics. However, for the complex solid–liquid coupling effects of the soft matter of hydrogels, the traditional single-phase wave model is no longer applicable. In this study, a transient wave propagation model that considers the solid–liquid coupling effects and its computational method is proposed. The wave-governing equations of hydrogels are derived based on the mass conservation and dynamic equilibrium equations, where the motions of solid and liquid phases are independently described. After transforming the governing equations into the equivalent weak form, numerical solutions for transient wave propagation in one- and two-dimensional hydrogels are calculated. The accuracy of the proposed model is verified by a comparison between semi-analytical and commercial finite element method solutions. We observe that the travelling and reflection processes of the two types of compression waves, P1 and P2, with different wave speeds are captured when the dynamic coefficient of permeability kf increases. Furthermore, the influence of solid–liquid coupling effects on the transient responses of hydrogels is discussed. The results show that the hydrogels exhibit the dynamic characteristics of a single-phase medium to a certain extent when kf is sufficiently small.

Bending analysis of thin plates with variable stiffness resting on elastic foundation via a two-network strategy physics-informed neural network method

A physics-informed neural network method based on a two-network strategy is introduced to address the bending problem of thin plates with variable stiffness resting on an elastic foundation. This problem is governed by a fourth-order partial differential equation (PDE), and the use of a one-network strategy to solve the PDE directly may lead to convergence issues due to singular points. Following the principles of Kirchhoff plate theory, the governing PDE is equivalently transformed into four second-order PDEs. A two-network strategy is employed for solution. We present numerical examples under various load conditions, plate geometries, foundation types, thicknesses, and material properties. The obtained results are validated against finite element method (FEM) solutions and literature. Discussions on alternative network strategies were conducted, and this study reveals that the presented two-network strategy have proven to be effective and robust. It also facilitates easier imposition of boundary conditions.

Theoretical and numerical analysis of end force of multilayer bellows expansion joint

Multilayer bellows expansion joint is a thin-walled shell widely used in underground pipeline systems for displacement compensation and vibration absorbing. The bellows expansion joint’s end force affects the underground pipeline’s safety. It is urgent to solve the end force of the bellows expansion joint theoretically. However, scholars only obtained the coupling solution of the end force of the bellows expansion joint, which is inconvenient in theoretical analysis and engineering application. This paper proposes a decoupling solution for a bellows expansion joint based on theoretical and numerical methods. The end force is decoupled to the deviation of the bending pressure which is caused by overcoming its own stiffness and internal pressure. In the end force expression, stiffness and internal pressure are independent of each other, which can evaluate the influence of asymmetric pressure difference and external load on end force more accurately. In the end, the accuracy of the decoupling solution is verified by the finite element method and coupling solution. The results show that the decoupling solution agrees with the results obtained by the finite element method and coupling solution. It can provide a theoretical basis for the design of the bellows structure.

Transfer learning‐based physics‐informed neural networks for magnetostatic field simulation with domain variations

AbstractPhysics‐informed neural networks (PINNs) provide a new class of mesh‐free methods for solving differential equations. However, due to their long training times, PINNs are currently not as competitive as established numerical methods. A promising approach to bridge this gap is transfer learning (TL), that is, reusing the weights and biases of readily trained neural network models to accelerate model training for new learning tasks. This work applies TL to improve the performance of PINNs in the context of magnetostatic field simulation, in particular to resolve boundary value problems with geometrical variations of the computational domain. The suggested TL workflow consists of three steps. (a) A numerical solution based on the finite element method (FEM). (b) A neural network that approximates the FEM solution using standard supervised learning. (c) A PINN initialized with the weights and biases of the pre‐trained neural network and further trained using the deep Ritz method. The FEM solution and its neural network‐based approximation refer to an computational domain of fixed geometry, while the PINN is trained for a geometrical variation of the domain. The TL workflow is first applied to Poisson's equation on different 2D domains and then to a 2D quadrupole magnet model. Comparisons against randomly initialized PINNs reveal that the performance of TL is ultimately dependent on the type of geometry variation considered, leading to significantly improved convergence rates and training times for some variations, but also to no improvement or even to performance deterioration in other cases.

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

A flux-based moving mesh method applied to solving the Poisson–Nernst–Planck equations

The moving mesh method is one of the important adaptive mesh methods which is practically useful when the mesh size and overall resolution are provided and fixed as seen in many engineering computations. We employ a moving mesh technique combined with a finite element method (FEM) to solve a widely-used charged carrier transport model, the Poisson–Nernst–Planck (PNP) equations, the solutions of which often have boundary or internal layers and sharp interfaces when the convection is dominated. Considering that flux is a significant physical quantity in designing stable and accurate numerical algorithm for transport problems such as the PNP system, we start from a relatively general functional and propose a flux-based monitor function and an adaptive step size-controlling algorithm for guiding the mesh movement in the FEM solution of the PNP equations. The numerical results illustrate that the moving mesh method is effectively addressing challenges arising from solution singularities and the convection-dominated effects. It also shows that the moving mesh finite element method we propose exhibits superior performance compared to both the traditional moving mesh finite element method and fixed mesh finite element method in some scenarios. Furthermore, it demonstrates better adherence to the physical properties of energy dissipation inherent in the PNP equation.

A semi-analytical simulation method for bi-directional functionally graded cantilever beams under arbitrary static loads

This paper presents a novel semi-analytical simulation approach for analysing the behaviour of bi-directional functionally graded cantilever beams subjected to arbitrary static loads, such as concentrated moments, concentrated forces, distributed force and their combinations applied at any location along the beam. The fundamental equations governing the cantilever beam’s response are derived, on the basis of which the proposed semi-analytical method is implemented using MATLAB programming language. The simulation results include field variables as well as stress contours, providing a compressive understanding of the beam’s behaviour. To validate the accuracy and reliability of the proposed method, a convergence study is conducted in comparison with the graded finite element method (GFEM) and analytical solutions. In the end, the developed method is applied to simulate the bending behaviour of bi-directional functionally graded cantilever beams under various loads individually and their combinations. The stress contours and deflection curves obtained from the simulation are compared with the solutions obtained using GFEM, revealing that the developed method possesses excellent capability in accurately simulating the bending behaviour of cantilever beams.

Variational three-field reduced order modeling for nearly incompressible materials

This study presents an innovative approach for developing a reduced-order model (ROM) tailored specifically for nearly incompressible materials at large deformations. The formulation relies on a three-field variational approach to capture the behavior of these materials. To construct the ROM, the full-scale model is initially solved using the finite element method (FEM), with snapshots of the displacement field being recorded and organized into a snapshot matrix. Subsequently, proper orthogonal decomposition is employed to extract dominant modes, forming a reduced basis for the ROM. Furthermore, we efficiently address the pressure and volumetric deformation fields by employing the k-means algorithm for clustering. A well-known three-field variational principle allows us to incorporate the clustered field variables into the ROM. To assess the performance of our proposed ROM, we conduct a comprehensive comparison of the ROM with and without clustering with the FEM solution. The results highlight the superiority of the ROM with pressure clustering, particularly when considering a limited number of modes, typically fewer than 10 displacement modes. Our findings are validated through two standard examples: one involving a block under compression and another featuring Cook’s membrane. In both cases, we achieve substantial improvements based on the three-field mixed approach. These compelling results underscore the effectiveness of our ROM approach, which accurately captures nearly incompressible material behavior while significantly reducing computational expenses.

An Eight-Node Non-Conforming Generalized Partial Hybrid Element and Its Application in Stress Analysis of Repaired Composite Laminate Structures

To ensure the overall continuity of displacement and out-of-plane stress in composite laminate structures and to quantitatively analyze the mechanical properties of composite materials after damage or repair, a finite element solution method is applied based on the modified generalized H–R variational principle. This method utilizes an eight-node non-conforming generalized partial hybrid element (NCGPME8). The partial hybrid model established with this hybrid element can accurately satisfy the out-of-plane stress boundary conditions of the structure, ensuring the continuity of out-of-plane stress. Numerical examples are used to validate that this hybrid model can effectively compute thick and thin laminate structures with high accuracy and rapid convergence of out-of-plane stress. Finally, considering the insensitivity to irregular meshes and the accuracy in calculating in-plane stress, this method is propagated by element coefficient deduction or element material replacement, then employed to analyze the in-plane and out-of-plane stress distributions of laminates with damage from stepwise grinding perforations, and laminates repaired in a stepwise fashion. Stress and displacement at different locations on the laminates are compared and analyzed, leading to a quantitative assessment of the impact of damage and repair on the stress distribution of the laminates.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Finite element modeling of the rotational dynamics of a single magnetic particle in a strong magnetic field and liquid medium

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles.

A new crack-tip element for the logarithmic stress-singularity of Mode-III cracks in spring interfaces

A new crack-tip finite element able to improve the accuracy of Finite Element Method (FEM) solutions for cracks growing along the Winkler-type spring interfaces between linear elastic adherents is proposed. The spring model for interface fracture, sometimes called Linear-Elastic (perfectly) Brittle Interface Model (LEBIM), can be used, e.g., to analyse fracture of adhesive joints with a thin adhesive layer. Recently an analytical expression for the asymptotic elastic solution with logarithmic stress-singularity at the interface crack tip considering spring-like interface behaviour under fracture Mode III was deduced by some of the authors. Based on this asymptotic solution, a special 5-node triangular crack-tip finite element is developed. The generated special singular shape functions reproduce the radial behaviour of the first main term and shadow terms of the asymptotic solution. This special element implemented in a FEM code written in Matlab has successfully passed various patch tests with spring boundary conditions. The new element allows to model cracks in spring interfaces without the need of using excessively refined FEM meshes, which is one of the current disadvantages in the use of LEBIM when stiff spring interfaces are considered. Numerical tests carried out by h-refinement of uniform meshes show that the new singular element consistently provides significantly more accurate results than the standard finite elements, especially for stiff interfaces, which could be relevant for practical applications minimizing computational costs. The new element can also be used to solve other problems with logarithmic stress-singularities.

Elasto-Static Analysis of Composite Restorations in a Molar Tooth: A Meshless Approach.

Dental caries and dental restorations possess a long history and over the years, many materials and methods have been invented. In recent decades, modern techniques and materials have brought complexity to this issue, which has created the necessity to investigate more and more to achieve durability, consistency, proper mechanical properties, efficiency, beauty, good colour, and reduced costs and time. Combined with the recent advances in the medical field, mechanical engineering plays a significant role in this topic. This work aims at studying the elasto-static response of a human molar tooth as a case study, respecting the integral property of the tooth and different composite materials of the dental restoration. The structural integrity of the case study will be assessed through advanced numerical modelling resorting to meshless methods within the stress analysis on the molar tooth under different loading conditions. In this regard, bruxism is considered as being one of the most important cases that cause damage and fracture in a human tooth. The obtained meshless methods results are compared to the finite element method (FEM) solution. The advantages and disadvantages of the analysed materials are identified, which could be used by the producers of the studied materials to improve their quality. On the other hand, a computational framework, as the one presented here, would assist the clinical practice and treatment decision (in accordance with each patient's characteristics).

Multi-Disease Detection Using a Prism-Based Surface Plasmon Resonance Sensor: A TMM and FEM Approach.

This research introduces a surface plasmon resonance (SPR)-based biosensor with multilayered structures for telecommunication wavelength in order to detect multiple diseases. The malaria and the chikungunya viruses are taken into account and the presence of these viruses are determined by examining several blood components in healthy and affected phases. Here, two distinct configurations (Al-BTO-Al-MoS2 and Cu-BTO-Cu-MoS2) are proposed and contrasted for the detection of numerous viruses. The performance characteristics of this work have been analyzed using Transfer Matrix Method (TMM) method and Finite Element Method (FEM) method under angle interrogation technique. From the TMM and FEM solutions, it is evident that the Al-BTO-Al-MoS2 structure provides the highest sensitivities of ~270 deg./RIU for malaria and ~262 deg./RIU for chikungunya viruses, with satisfactory detection accuracy of ~1.10 for malaria, ~1.64 for chikungunya, and quality factor of ~204.40 for malaria, ~208.20 for chikungunya. In addition, the Cu-BTO-Cu MoS2 structure offers the highest sensitivities of ~310 deg./RIU for malaria and ~298 deg./RIU for chikungunya, with satisfactory detection accuracy of ~0.40 for malaria, ~0.58 for chikungunya, and quality factor of ~89.85 for malaria, ~86.38 for chikungunya viruses. Therefore, the performance of the proposed sensors is analyzed using two distinct methods and gives around similar results. In a sum, this research could be utilized as a theoretical foundation and first step in the development of a real sensor.