Finite Element Solution Research Articles

A plastic correction algorithm for full-field elasto-plastic finite element simulations: critical assessment of predictive capabilities and improvement by machine learning

AbstractThis paper introduces a new local plastic correction algorithm that is aimed at accelerating elasto-plastic finite element (FE) simulations for structural problems exhibiting localised plasticity (around e.g. notches, geometrical defects). The proposed method belongs to the category of generalised multi-axial Neuber-type methods, which process the results of an elastic prediction point-wise in order to calculate an approximation of the full elasto-plastic solution. The proposed algorithm relies on a rule of local proportionality, which, in the context of J2 plasticity, allows us to express the plastic correction problem in terms of the amplitude of the full mechanical tensors only. This lightweight correction problem can be solved for numerically using a fully implicit time integrator that shares similarities with the radial return algorithm. The numerical capabilities of the proposed algorithm are demonstrated for a notched structure and a specimen containing a distribution of spherical pores, subjected to monotonic and cyclic loading. As a second point of innovation, we show that the proposed local plastic correction algorithm can be further accelerated by employing a simple meta-modelling strategy, with virtually no added errors. At last, we develop and investigate the merits of a deep-learning-based corrective layer designed to reduce the approximation error of the plastic corrector. A convolutional architecture is used to analyse the neighbourhoods of material points and outputs a scalar correction to the point-wise Neuber-type predictions. This optional brick of the proposed plastic correction methodology relies on the availability of a set of full elasto-plastic finite element solutions to be used as a training data-set.

Anisotropic error estimator for the Stokes–Biot system

This paper presents an a posteriori error analysis for the problem defining the interaction between a free fluid and poroelastic structure approximated by finite element methods on anisotropic meshes in Rd, d=2 or 3. Korn’s inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to nonconforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for conforming and nonconforming cases. With the obtained finite element solutions, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the reliability, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media.

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

A Gauss kernel non-local stress-driven plate theory

This study presents a novel stress-driven formulation, incorporating Kirchhoff’s plate assumptions and accounting for the non-local size-dependent feature in both plate geometric axes, which is practical, especially near the origin of the axis in the polar coordinates. The presented formulation does not impose any additional constraints on the elastic radial curvatures, which are known as constitutive boundary conditions. Therefore, taking into account the non-local behavior in both radial and circumferential directions does not lead to an overconstrained problem. The study focuses on the analysis of a circular plate with an outer radius (R), thickness (h), and the length scale parameter (L). The governing differential equations are derived in Cartesian coordinates and then transformed to the polar coordinates to analyze circular plates. When R tends to infinity, the governing equation is solved analytically. Using the Galerkin technique, a finite element solution is presented for the analysis of a bounded circular plate with a clamped boundary condition at the outer radius R. The study discusses the size effects and the gradient properties of functionally graded plates in stress-driven non-local theory. It reveals that while increasing the non-local parameter results in a reduction of transverse displacement, the moments remain unaffected.

Peridynamics model of torsion-warping: Application to lattice beam structures

This paper presents a finite deformation beam model based on Simo-Reissner theory in peridynamics (PD) framework to deal with torsion induced warping deformation. Seven degrees of freedom, viz. three translational, three rotational, and one warping amplitude are considered at each material point. The governing equations of the beam are obtained by employing global balance of linear and angular momenta in conjunction with Simo's assumption on the deformation field. The relation between PD resultant force, moment, bi-moment, and bi-shear states with their classical counterparts is established using the constitutive correspondence method. Numerical implementation strategy is furnished for both quasi-static and dynamic cases. The solution for quasi-static load is obtained through the Newton-Raphson method. The proposed model is validated against finite element solutions considering cantilever beam and lattice structures. Quasi-static deformation responses of 3×3×3 octet and single unit compression-torsion lattice structures are presented further to demonstrate the effectiveness of proposed beam model. A new bond breaking criterion is proposed based on critical stretch, critical relative rotation, and critical relative warping amplitude and failure of the compression-torsion lattice structures under compressive load is simulated. The Newmark-beta method is utilized to solve the governing equations for dynamic loading. Numerical simulations include dynamic analysis of octet and compression-torsion lattice structures.

New Analytical Solutions for Constant Rate Pumping in Two‐Zone Double‐Porosity Confined Aquifer: A New Source Term Reflecting Effects of Well Skin and Wellbore Storage

AbstractThis study develops two new analytical models for constant rate pumping at a partially penetrating well in a double‐porosity confined aquifer, considering skin and formation zones. One model, referred to as the two‐zone model, incorporates a flow equation to depict the flow in the skin around the well. The other model, named the source‐term model, introduces a novel source term at the outer rim of the skin to reflect the effects of both the skin and wellbore storage. The analytical solutions for both models are derived by the Laplace transform and finite Fourier cosine transform. Additionally, a finite element solution for the source‐term model is presented. Results suggest the source‐term model is suitable to most wells when the width of the skin is less than 1 m and the radius of influence exceeds the outer rim of the skin. Temporal drawdown distribution for a negative skin exhibits a triple‐humped shape with two flat stages, while that for a positive skin shows monotonous increase. The source‐term model enables orthogonal 5 × 5 nodes for finite element approximation to discretize a well and its adjacent skin. The finite element solution aligns with early drawdown data measured at an observation well under the effects observed in two field constant rate pumping tests. In conclusion, this study introduces a novel approach to modeling two‐zone flow, which may find practical utility in field applications.

A novel post-processed finite element method and its convergence for partial differential equations

In this article, by combining high-order interpolation on coarse meshes and low-order finite element solutions on fine meshes, we propose a novel approach to improve the accuracy of the finite element method. The new method is in general suitable for most partial differential equations. For simplicity, we use the second-order elliptic problem as an example to show how the novel approach improves the accuracy of the finite element method. Numerical tests are also conducted to validate the main theoretical results.

Effects of Glacial Isostatic Adjustment on Fault Reactivation and Its Consequences on Radionuclide Migration in Crystalline Host Rocks

AbstractTo assess the robustness of a safety case for a deep geological repository (DGR), it is necessary to analyze a range of scenarios covering likely, less likely, and hypothetical future developments. Crystalline rock can, under ideal conditions, provide a suitable hydrogeologic barrier due to its extremely low matrix permeability. However, this host rock is often fractured, which can compromise its hydro-mechanical (HM) barrier function. We quantify how faults that are prone to reactivation during glacial events can affect radionuclide migration around a DGR in a crystalline host rock. We extend a previously developed finite element model of coupled fluid flow and radionuclide transport to numerically solve the component transport problem before and after fault reactivation. Assuming that fault reactivation is triggered by changes in mechanical boundary conditions, we derive heterogeneous permeability distributions in the reactivated faults by evaluating the Coulomb failure stress criterion of finite element solutions of a complementary hydro-mechanical problem. Specifically, we evaluate the consequences of glacial isostatic adjustment (GIA) during a glacial cycle. We find that the increased permeability in the reactivated faults accelerates the migration of radionuclides along the fault by channeling the flow, while it is reduced in the direction perpendicular to the fault. The channeling observed is also a result of heterogeneous permeability enhancement, and the flow fields differ from those of the previous model which postulated a homogeneous permeability enhancement. Although the proposed numerical workflow has been applied to the case of GIA, it is adaptable to study hydro-mechanical processes induced by seismic events or by hydrofracking in enhanced geothermal systems.

Optimal lamination angles via exact and efficient differentiation of the geometrically nonlinear finite element solution

Laminates allow tailoring the fiber orientations in the layers to obtain the desired mechanical response. The optimal layup design is a challenging task in the case of finite deformations and buckling. For an assigned design, an incremental-iterative finite element analysis is needed to compute the structural response. Gradient-based methods are very often the most efficient optimization tools. Their bottleneck is the gradient evaluation, generally possible only approximately by finite differences. This article shows how to compute the exact gradient of the geometrically nonlinear finite element solution with respect to the stacking sequence. The strategy relies on the implicit differentiation of the nonlinear discrete equations of a control equilibrium point corresponding to an assigned displacement. This provides the load factor gradient by a single fast-solution linear system with the already factorized tangent stiffness matrix, regardless of the number of design variables, and scalar products involving the partial derivatives of the discrete internal force vector, calculated in an exact and efficient way. Several applications demonstrate the efficiency and robustness of the approach.

Multi-helix buckling of a slender compression rod constrained by a cylinder

As early as the mid-18th century, Euler proposed a critical load formula for compression rods with multiple half-sine wave modes. However, a critical load formula for compression rods constrained by cylinders with multi-helix buckling modes has not yet been proposed. In this paper, a mechanical model of multi-helix buckling of a slender compression constrained by a cylinder is established. Utilizing the dynamic relaxation method, a finite element solution strategy is presented. Its notable advantage lies in its capability to compute stable configurations of multiple helical buckling, effectively addressing challenges such as contact identification and convergence issues during computation. Furthermore, parameter analysis quantifies the influence of length, diameter, and annular gap on the critical load, while also analyzes the shear force, bending moment, and contact force after buckling. This paper presents formulas for the critical load of multi-helix buckling and the length of the multi-helix buckling modes, which is suitable for various length, diameter, and annular space gap conditions, and partially validates these results against finite element analysis.

Methodology for the thermoelastic stress analysis of complex biomimetic structures: An alligator mandible case study

Biomimetic designs that take inspiration from natural structures provide a unique approach to identifying innovative solutions to engineering problems. The experimental analysis of such structures, however, can be difficult due to complex, three-dimensional geometric features. This study presents a methodology for the analysis of complex biomimetic structures using thermoelastic stress analysis (TSA) illustrated through an alligator mandible case study. This structure has evolved to become optimal for its function and environment which has resulted in unique and intricate features. An additively manufactured alligator mandible structure was experimentally analysed using TSA under bending load conditions and compared to a finite element (FE) solution. TSA was found to be a suitable method for the qualitative and quantitative analysis of the complex biomimetic structure which correlated well with the FE model. This study demonstrates the accuracy of TSA for the assessment of biomimetic designs with unique, arbitrary geometric features that can be utilised for future nature-inspired engineering applications.

Convergence and superconvergence analysis for a mass conservative, energy stable and linearized BDF2 scheme of the Poisson–Nernst–Planck equations

In this paper, we consider a linearized BDF2 finite element scheme for the Poisson–Nernst–Planck (PNP) equations. By employing a novel approach, we rigorously derive unconditional optimal error estimates of the numerical solutions in the l∞(L2) and l∞(H1) norms, as well as superconvergent results. The key of the convergence and superconvergence analysis lies in deriving the stability of the finite element solutions in some stronger norms. The advantage of this approach is that there is no need to introduce a corresponding time discrete system, so it is more concise than the error split technique in previous literatures. Finally, we carry out two numerical examples to confirm the theoretical findings.

The dynamic performance of semi-infinite piezoelectric strip containing irregular boundaries based on the boundary element method

In this article, the problem of a semi-infinite piezoelectric strip with irregular boundaries is studied by boundary element method, and SH guided wave and line source loads are considered as external loads acting on piezoelectric strip. The superiority of the boundary element method is induced by two different arithmetic examples. Firstly, when the irregular boundary is not subjected to line source loads, the dynamic characteristics are analyzed in the first example, employing the image method and Graf addition theorem. Then, the Green’s identities are introduced and the Green’s function in infinite three-dimensional space is solved. When irregular boundaries are subjected to line source loads, the dynamic characteristics are investigated using the boundary element method in the second example. Finally, results clarified the influence on the dynamic stress concentration factor and electric field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

Analytical solutions for static longitudinal displacements of suspension bridges under a moving vertical concentrated load

The longitudinal displacement at the girder ends of long-span suspension bridges is a crucial issue that affects structural safety, durability, as well as traffic safety and ride comfort. The primary cause of these reciprocating displacements is the action of moving live loads, which induces quasi-static displacements in the longitudinal direction. This paper theoretically investigates the static longitudinal displacements of suspension bridges under a moving vertical concentrated load and elucidates the underlying deformation mechanisms. Considering geometrical nonlinearity, analytical equations are established for the longitudinal deformations of the main cable and stiffening girder under a vertical concentrated load. By analyzing the geometrical deformation conditions of the suspenders, the relationship between the longitudinal displacements of the stiffening girder and the main cable is derived. Due to the coupling between longitudinal and vertical displacements, the vertical displacement of the stiffening girder must be solved to obtain its longitudinal displacement. Based on deflection theory and considering the bending stiffness of the stiffening girder, this paper derives the vertical deformations of the stiffening girder for single-span suspension bridges and those with short continuous side spans under a vertical concentrated load, which is then used to calculate the longitudinal displacement. Finite element models are used to verify the accuracy of the analytical solutions. Differences between the proposed solutions and previous solutions neglecting the bending stiffness of the stiffening girder are compared and analyzed. The effects of neglecting the longitudinal displacement at the bridge tower top and the presence of short continuous side spans on the longitudinal displacement of stiffening girder are also investigated. The results demonstrate the high accuracy of the proposed analytical solution for the longitudinal displacement at the girder ends of suspension bridges. Importantly, the bending stiffness of the stiffening girder cannot be neglected in calculating the longitudinal displacement. The longitudinal displacement at the girder ends is primarily caused by the longitudinal deformation of the main cable, while the contribution from the flexural deformation of the girder is negligible. Furthermore, the longitudinal displacement at the bridge tower top has a negligible effect on the longitudinal displacement of the girder, and the presence of short continuous side spans is beneficial for reducing the longitudinal displacement of the stiffening girder. The proposed analytical solution method provides a rapid approach for calculating the static longitudinal displacement at the girder ends of suspension bridges under passing live loads and reveals the deformation mechanism. This is beneficial for preliminary design and structural optimization, as it avoids the complex finite element modeling and solution process.

Finite deformation peridynamics shell theory: Application to mechanical metasurfaces

A finite deformation Simo-Reissner peridynamics (PD) shell theory is developed to investigate wave propagation and crack growth in shell structures. Simo-Reissner deformation approximation is utilized to derive the governing equations of the proposed theory from the three-dimensional PD equation. Deformation states for the PD shell theory are identified from the reduced stress power equation. Classical shell constitutive equations are seamlessly incorporated in PD framework by constitutive correspondence approach where a particular form of the classical reduced stress power expression suitable for constitutive correspondence is used. Rotation tensor is updated using an exponential map. A new bond breaking criterion is proposed for PD shell based on critical stretch and critical relative rotation. The solutions of the PD shell governing equations for quasi-static and dynamic cases are obtained through the Newton-Raphson method and the Newmark-beta method, respectively. Numerical simulations include finite deformation of cylindrical shell under both quasi-static and dynamic loading conditions. The PD shell model is validated against finite element solutions obtained using ANSYS®. Crack propagation through the cylindrical shell is simulated under quasi-static and dynamic loading. Wave propagation through cylindrical shell embedded with periodically arranged holes and cracks is investigated. Significant wave attenuation and large bandgap demonstrates the efficacy of the present proposal.

Analytical and FE modeling of a bi-directional functionally graded Timoshenko beam under thermomechanical loading

Functionally graded materials (FGMs) have variable volume fractions in different directions, making structural analysis challenging due to shear deformation. This study investigates the behavior of a two-directional functionally graded material (TDFGM) Timoshenko beam under thermomechanical loads. Varied material properties in both axial and transverse directions are analyzed using a mathematical approach and a finite element model. The mathematical approach involves transforming the system of two ordinary differential equations to a single fourth-order differential equation to determine deflection and rotation. Additionally, a bilinear quadrilateral element is used to validate the mathematical solution. The results show that increasing grading parameters reduced transverse deflection, with a maximum reduction of 38.97% for CF beams and 39.26% for SS beams when the gradation index az increased from 0 to 1 under mechanical loading. Under combined mechanical and thermal loading, higher temperatures on the beam’s hot side lead to increased deflection, with notable increases at higher gradation indices. Moreover, increased maximum deflection by 6.29% for CF beams and 4.05% for SS beams when considering temperature-dependent properties. It was observed that when TDFGM Timoshenko beams were heated for thermal loading, the SS beam deflected in the opposite direction of the applied mechanical load. The results agree with previous research and indicate alignment between mathematical and finite element solutions. The impact of boundary conditions on the deflection of Timoshenko beams made of TDFGM is discussed. The introduced methodology enhances TDFGM beam bending analysis, allowing control over behavior through grading parameters and boundary conditions selection.

Surface acoustic waves in porous soils with two-layered infiltration

Abstract Two stage mathematical and computer models based on the finite element solution of Richards’ equation for water infiltration in two-layered fluid-saturated soil and modal analysis using a semi-analytical approach for vertically stratified soils are presented and briefly described. Dispersion curves of surface waves in the layered poroelastic half-space are computed and studied. The features of porous water infiltration effect on the characteristics of surface elastic waves are discussed.

Bridging micro nature with macro behaviors for granular thermal mechanics

The connection between micro-level characteristics and macroscopic properties in granular heat transfer and mechanics is fundamental and crucial. This study proposes a novel discrete element approach incorporating granular heat transfer, contact bonding, and granular stress tensor models to investigate the mechanical and thermal responses of continuum media composed of constituent spheres. Eight benchmark tests were devised to bridge the long-standing gap between micro and macro properties in granular materials. Through these tests, the numerical solutions obtained from discrete element modeling match well with existing analytical or finite element solutions derived from continuum-based theory. This validation underscores the rationality and reliability of the granular heat transfer model, contact bonding model, and granular stress tensor model. Moreover, the study highlights the consistency between continuum-based theory and discontinuum-based theory. A minor distinction between continuum-based models and discrete element models emerges near the boundaries due to variations in the specification of boundary conditions. This discrepancy can be clarified by Saint-Venant's Principle, thus validating the accuracy of the microscale heat transfer and mechanics theory for granular materials. Five mono-disperse packing structures, including simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close packing (HCP), and random packing (Random), were further analyzed to examine their influence on heat transfer performance. Numerical results reveal that higher coordination numbers and solid volume fractions correspond to higher apparent thermal conductivity of granular assemblies, thus elucidating the connection between micro packing configurations and macroscopic heat transfer properties. The apparent thermal conductivity for different crystal configurations follows the sequence: HCP ≒ FCC > BCC ≒ Random > SC. To improve the accuracy and physical relevance of the proposed model, the effect of particle contact area needs to be further incorporated into the granular heat transfer model.

Discrete strong extremum principles for finite element solutions of advection‐diffusion problems with nonlinear corrections

AbstractA nonlinear correction technique for finite element methods of advection‐diffusion problems on general triangular meshes is introduced. The classic linear finite element method is modified, and the resulting scheme satisfies discrete strong extremum principle unconditionally, which means that it is unnecessary to impose the well‐known restrictions on diffusion coefficients and geometry of mesh‐cell (e.g., “acute angle” condition), and we need not to perform upwind treatment on the advection term separately. Moreover, numerical example shows that when a discrete scheme does not satisfy the strong extremum principle, even if it maintains the global physical bound, non‐physical numerical oscillations may still occur within local regions where no numerical result is beyond the physical bound. Thus, it is worth to point out that our new nonlinear finite element scheme can avoid non‐physical oscillations around sharp layers in advection‐dominate regions, due to maintaining discrete strong extremum principle. Convergence rates are verified by numerical tests for both diffusion‐dominate and advection‐dominate problems.