Finite Element Formulation Research Articles

The effects of nonlinear soil spatial variability on the nonlinear beam response


 
 
 
 The nonlinear analysis of a beam resting on nonlinear random tensionless soil was studied with the aim of quantifying the influence of the spatial variability of the tension soil characteristics on the behavior of the beam and illustrating the importance of the geometric nonlinear analysis of a beam. The soil–structure interaction mechanism is taken into account where the soil is modeled as nonlinear. Due to large deflections and moderate rotations of the beam, the Von-Kàrman type nonlinearity based on the finite element formulation of nonlinear beam response is adopted and the interfacial shearing resistance along the beam is taken into account. To assess the probabilistic nonlinear behavior of the beam resting on nonlinear random soil, a Monte Carlo approach was used and the response of the beam was statistically analyzed for different beam and soil parameters. The equation of motion is solved using the Newton-Raphson iterative method which is implemented by using a Matlab software. The results indicate that the probabilistic approach of the soil and the geometric nonlinearity of the beam serve a major role in the evaluations of the beam response (of the real response of the buried beam).
 
 
 

Modeling vortex-induced vibrations of branched structures by coupling a 3D-corotational frame finite element formulation with wake-oscillators

Branched structures are present in a diverse set of problems, from modeling branch pipe connections to simulating tree dynamics. Soft corals like the Bipinnate sea plume, have a branched geometry and are soft enough to bend under the waves. Due to their circular cross section, a vortex street forms in the coral’s wake inducing vibrations of its branches. Despite extensive studies on VIV in straight geometries, the three-dimensional (3D) dynamics of flexible branched structures remains uninvestigated. In this numerical and experimental study, we develop a novel formulation for the accurate computation of in-line and cross-flow VIV of frame structures undergoing large deformation. The finite element approach is used to model arbitrarily complex geometries of branched frame structures. Our formulation allows to model complex geometries with forks or sharp angles. The consistent 3D corotational formulation for frame elements computes the internal, inertial and hydrodynamic forces. A wake-oscillator approach models the near wake dynamics with fluctuating fluid forces on the structure in the in-line and cross-flow directions. The drag and lift coefficients follow distributed Van der Pol oscillators. Moreover, we implement the numerical resolution procedure in the open-source library ONSAS. The present formulation and numerical resolution procedure is validated by solving three examples, including comparisons with an analytical solution, a wake-oscillator, and experimental data from the literature. We also conduct experiments of a flexible and elastic cylinder clamped inside a water tunnel under a constant uniform flow. Amplitudes and power spectral density of the tip transverse displacements are compared with the model prediction. Finally, the proposed formulation is applied on a cylinder with two branches. The simulations demonstrate a multi-frequency response with higher amplitudes of displacements when additional branches are incorporated onto the cylinder, emphasizing the significance of considering VIV in nature and engineering applications for such geometries.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

An enhanced corotational Virtual Element Method for large displacements in plane elasticity

An enhanced virtual element formulation for large displacement analyses is presented. Relying on the corotational approach, the nonlinear geometric effects are introduced by assuming nodal large displacements but small strains in the element. The element deformable behavior is analyzed with reference to the local system, corotating with the element during its motion. Then, the large displacement-induced nonlinearity is accounted for through the transformation matrices relating the local and global quantities. At the local level, the Virtual Element Method is adopted, proposing an enhanced procedure for strain interpolation within the element. The reliability of the proposed approach is explored through several benchmark tests by comparing the results with those evaluated by standard virtual elements, finite element formulations, and analytical solutions. The results prove that: (i) the corotational formulation can be efficiently used within the virtual element framework to account for geometric nonlinearity in the presence of large displacements and small strains; (ii) the adoption of enhanced polynomial approximation for the strain field in the virtual element avoids, in many cases, the need for ad-hoc stabilization procedures also in the nonlinear geometric framework.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

A user material approach for the solution of multi-field problems in Abaqus: Theoretical foundations, gradient-enhanced damage mechanics and thermo-mechanical coupling

The solution of multi-field problems and the numerical implementation by means of the finite element method constitute a sophisticated part of the characterisation of industrial processes. A comprehensive implementation framework for such a system of coupled field equations into a non-linear large strain finite element formulation is provided. The procedure is derived for a micromorphic approach in a thermo-mechanical setting. Although the provided framework contributes to a particular three-field problem it is not limited to a specific application or a specific number of coupled field equations from a conceptual point of view. The solution of the considered system of equations is separated into two coupled domains, with the balance of linear momentum and a balance equation of heat equation-type being solved on each of them. Since both, the balance of micromorphic momentum and the heat balance equation, are partial differential equations of Laplace-type, the resulting two-instance problem can be solved in the framework of commercial finite element software, such as Abaqus, based on a thermo-mechanical user material. To assess the framework for a particular constitutive model, a gradient-enhanced damage model in a thermo-mechanical setting is applied and representative simulation results are discussed. The Abaqus framework is made available as an open-source code on GitHub (https://github.com/InstituteOfMechanics/Thermomechanical_Gradient_Enhanced_Damage_UMAT).

Free vibration analysis of hygrothermally stable stiffened composite plates with plate-stiffener interfacial debonding

Stiffeners are frequently utilized in many engineering projects because they boost strength while adding only a small amount of weight to the overall structure. The environments that the stiffened composite constructions are typically exposed to are harsh, which can result in damage like interlayer delamination and debond at the plate-stiffener interface, which can cause catastrophic failure. As a result, it is necessary to examine these structures’ dynamic behavior in addition to their stability performance. The current study attempts to investigate the free vibration response of stiffened composite plates with plate-stiffener interfacial debonding under hygrothermal circumstances using a robust finite element (FE) model. The skin and stiffener flange are simulated in the current work using a 9-noded heterosis element to prevent the shear-locking issue. Additionally, to considerably enhance computational performance, the stiffener web is formulated as a 3-noded isoparametric beam element with the torsion correction factor. Additionally, a dummy node is created to represent the debond, and a fictitious spring is added to stop the nodes from interpenetrating one another. Three schemes of hygrothermally stable laminates (θ/(90- θ))s, (22.5/−67.5)s and (77.5/−12.5)s are considered to identify the stiffened plate configuration with improved free vibration characteristics. Further, extensive parametric analyses are conducted on the obtained stiffened plate with improved performance in a hygrothermal environment by considering debond at the skin-stiffener interface. The vibrational behavior of the debonded plate is shown to be significantly more affected by moisture than by temperature. Additionally, the debond has a more obvious impact on the vibration characteristics in the plate with a centrally attached stiffener than in the plate with an eccentricity attached stiffener. As a result, the established FE formulation is anticipated to be trustworthy and computationally effective while effectively analyzing the debonded stiffened panel’s free vibration behavior in a hygrothermal environment.

A stabilized finite element method for modeling dispersed multiphase flows using orthogonal subgrid scales

We propose a finite-element formulation for simulating multi-component flows occupying the same domain with spatially varying concentrations. Each constituent is assumed to behave as an incompressible Newtonian fluid, and solutions are sought for the velocities and volume fractions of each phase, as well as the common pressure. Stabilization terms are derived within the framework of the variational multiscale method based on an approximation of the finite-element residual to achieve control of the pressure and volume fractions. We utilize the concept of term-by-term stabilization in conjunction with orthogonal subgrid scales, thus incorporating only those terms of the residual essential to obtain stability and projecting them on a space orthogonal to the finite element space. The resulting system of equations is solved in a monolithic manner, requiring a small number of nonlinear iterations. Several benchmark tests have been performed to confirm the stability and optimal asymptotic convergence rates for linear and higher-order elements using the proposed formulation.

A new C0 continuous refined zigzag {1,2} finite element formulation for flexural and free vibration analyses of laminated composite beams

This study presents a novel C0 continuous refined zigzag finite element formulation {1,2}, namely RZE{1,2}, for the bending and free vibration analyses of laminated composite beams. The Refined Zigzag Theory (RZT) effectively combines accuracy and computational efficiency, making it a robust approach for thin and thick laminated composite structures. The RZT eliminates the need for shear correction factors, thereby enhancing the overall streamline of the analysis process. The present RZE{1,2} formulation takes into account the transverse stretching by introducing quadratic through-thickness variations of deflection components. The governing equations of the RZT are derived by means of Hamilton’s principle. The through-the-thickness variations of the transverse shear and normal stresses are calculated by integrating the stress equilibrium equations in a post-processing step. Therefore, the Peridynamic Least Squares Minimization (PDLSM) approach is utilized to obtain precise derivatives of axial stresses in the stress equilibrium equations. A new finite element formulation is built up with 3-nodes with a total of 15 DOF. In order to study the influence of transverse stretching, different types of boundary conditions and material variations are applied for laminated composite beams for the bending and free vibration analyses. The distribution of displacements as well as the axial and transverse stresses of the thick beams are extensively examined. The outcomes of RZT are in good agreement with the reference solutions in the literature. The findings of the present study reveal that the presence of the transverse stretching produces more realistic predictions for thick beams where the shear deformations are influential.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

State of art for hybrid mixed finite element formulation in non-linear analysis of structures

Since the late 80’s the Structural Analysis Research Group of the Instituto Superior Técnico (IST) has been involved in the development of non-conventional finite element formulations in order to overcome some of the limitations associated with the use of the CFE method and to develop high performance numerical tools for the analysis of structural engineering problems. Several alternative models for the linear and non-linear structural analysis have been developed using hybrid and mixed models techniques. These works are summarized in this paper, in which their past and future applications of this formulation in non-linear analysis of structures are fully detailed.

On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

Analysis of aWeak Galerkin Mixed Formulation for Maxwell’s Equations

In this paper we introduce and analyse a mixed weak Galekin finite element method for the Maxwell equations in the primary electric field-Lagrange multiplier. Our weak Galerkin method is equipped with stable finite elements composed of habitual polynomials of degree k for the electric field and polynomials of degree k + 1 for the Lagrange multiplier. Optimal order error estimations for the proposed weak Galerkin mixed finite element formulation are demonstrated and are confirmed numerically on a two dimensional bounded domain.

An enhanced SUPG-stabilized finite element formulation for simulating natural phenomena governed by coupled system of reaction-convection-diffusion equations

Many phenomena arising in nature, science, and industry can be modeled by a coupled system of reaction-convection-diffusion (RCD) equations. Unfortunately, obtaining analytical solutions to RCD systems is typically not possible and, therefore, usually requires the use of numerical methods. On the other hand, since solutions to RCD-type equations can exhibit rapid changes and may have boundary/inner layers, classical computational tools yield approximations polluted with physically meaningless oscillations when convection dominates the transport process. Towards that end, in order to eliminate such numerical instabilities without sacrificing accuracy, this work employs a stabilized finite element formulation, the so-called streamline-upwind/Petrov-Galerkin (SUPG) method. The SUPG-stabilized formulation is then also supplemented with the YZ$\beta$ shock-capturing mechanism to achieve higher-quality approximations around sharp gradients. A comprehensive set of numerical test experiments, including cross-diffusion systems, the Schnakenberg reaction model, and mussel-algae interactions, is considered to reveal the robustness of the proposed formulation, which we call the SUPG-YZ$\beta$ formulation. Comparisons with reported studies reveal that the proposed formulation performs quite well without introducing excessive numerical dissipation.

Wave propagation in beams with functionally graded porosity distribution under highly transient axial and transverse impacts

Recent advances in the manufacturing process provide a possibility of fabricating a new generation of porous materials denoted by functionally graded porous materials (FGPM). This paper aims to present a time domain analysis of wave propagation through the porous structures with functionally graded porosity distribution, which has not been completely studied before. For this purpose, the beams with different functionally graded porosity distributions subjected to both axial and transverse tip impact loads with a high-frequency content are investigated. The shear deformable cantilevered functionally graded porous beams with various porosity distributions through the beam thickness are studied. The governing differential equations are derived using the Hamiltonian principle based on the Timoshenko beam theory. A locking-free first-order shear deformable beam element is used to derive the finite element formulation of the equations. The Newmark time integration method is used to perform a time domain analysis of the equations of motion and to investigate the transient response of the beams. The axial and transverse wave propagation characteristics through functionally graded (FG) porous beams are found using time domain analysis of the results. Deflection and velocity time histories of the tip and each point of the beam, reflection time, and variation of support reactions are obtained. The influences of the porosity magnitude and porosity distribution on the wave propagation characteristics and overall time responses are investigated. The results reveal that porosity distribution has a significant effect on the wave amplitude, wave speed, and reflection from the boundary. Also, this study can help in a better understanding of porous structures' behavior subjected to high-transient impact loads in different engineering applications.

A two-stage based model for partially-buried pile group response in cross-anisotropic saturated media adjacent to moving harmonic loads

The pile foundations are frequently affected by adjacent traffic loads except for general active loads. However, the dynamic responses of the pile groups under moving loads have been rarely reported before. In addition, the influence of material anisotropy is often neglected. In this paper, a three-dimensional (3D) analytical model for the dynamic analysis of partially buried pile groups in stratified saturated cross-anisotropic media under adjacent moving harmonic loadings is developed. Specifically, a 3D Bernoulli–Euler beam theory is adopted to simulate the monopile and then superimposed into the finite element (FE) formulation for the partially embedded pile groups. Subsequently, the basic solutions of layered saturated cross-anisotropic soils are derived by the analytical layer-element approach (ALEA). By combination with ALEA, the two-stage based boundary element (BE) equations for the pile-soil interface are obtained. Finally, by coupling the FE and BE formulations, the dynamical response equation of pile-soil is established. Based on the verification of the accuracy of the proposed methodology by comparisons with existing solutions and FE results from ABAQUS, the impacts of material anisotropy, pile buried length, loading velocity, and pile stiffness on the time-domain dynamic behavior of pile groups are analyzed. The results show that increasing the pile-buried ratio or pile stiffness reduces the magnitude of the dynamic response. Moreover, with the increase of material anisotropic parameters, the peak pile response decreases. And the peak dynamic response first increases and then reduces as the loading velocity increases.

Investigation of the edge effect in composites using the ANSYS software in a two-dimensional formulation

The possibility of using the universal software ANSYS to study the concentration of interlaminar stresses near the free edges of laminated composites in a two-dimensional finite element formulation is considered. In this case, the cross-section of a rectangular four-layer composite plate of finite width with free side edges subject to uniform axial deformation is simulated. To create it, an original technique is proposed which consists in using thermal stress. The presented calculation results are in good agreement with the results of other authors.

A CutFEM method for phase change problems with natural convection

In this paper, we present a Nitsche-based CutFEM approach for solving phase change problems while taking natural convection into account in the liquid phase. In the proposed algorithm, the interface is implicitly tracked through a level-set approach, where the level-set function is updated by solving a transport equation. To stabilize the finite element formulation of this equation, we use the continuous interior penalty method (CIP). We use the Backward Euler scheme for the time discretization and the of continuous piecewise linear polynomials (P1) for the spatial discretization of the temperature and the level-set function. For flow velocity and pressure in the liquid phase, the finite element formulation is based on the inf-sup stable pair of Hood-Taylor finite element function spaces P2−P1. As we are dealing with unfitted finite element methods, stability issues may arise due to geometry discretization. To address this issue, we incorporate ghost penalty terms into the finite element formulations of the heat and Navier–Stokes equations. According to Stefan’s condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. For an efficient approximation of this jump, we use a ghost penalty-based domain integral approach which has shown to be highly accurate. To validate the proposed method, we first provide two manufactured solutions that confirm optimal convergence in both temporal and spatial discretizations. We next consider challenging problems such as the melting of n-octadecane in a differentially heated cavity, the multi-cellular melting of gallium, and the freezing of water to further validate our algorithm. We compare our results to existing numerical and experimental data.

Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates

In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and C. elegans.In this work, we developed a computationally efficient 3D Finite Element (FE) unified framework for the locomotion of limbless organisms on frictional substrates. Muscle activity is simulated with a multiplicative decomposition of the deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D contractile elastic solids.We propose a two-field FE formulation based on positions and velocities. Governing partial differential equations are transformed into equivalent time-continuous differential–algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure is discussed, and Symplectic Euler time integration of optimality conditions are employed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backward Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal strategy out of distinct locomotion patterns adopted by limbless organisms.

Performance of 2D-spectral finite element method in dynamic analysis of concrete gravity dams

This article presents a two-dimensional spectral finite element formulation for dynamic analysis of concrete gravity dams. Finite element method (FEM) is the one of the most extensively used analysis tools for solving problems of dynamic analysis of structures. However, it needs extensive computational resources and time for large structures, leading to the development of alternate computationally efficient modeling techniques over the past few decades. Some of these techniques are broadly termed as “Spectral Finite Element Methods” (SFEMs). Frequency domain-based SFEM (FDSFEM) and time domain-based SFEM (TDSFEM) are the most commonly used SFEMs found in the literature. This article provides a brief review of these methods identifying their key advantages and disadvantages; and explores the feasibility of applying such methods in the dynamic analysis of concrete gravity dams. Here, the TDSFEM has been used to perform the dynamic time history analysis of a concrete gravity dam due to its relative advantages over FDSFEM, which is challenging for irregular geometry and finite domain. Sensitivity analysis and convergence studies have been performed using 4-noded and 9-noded elements. The foundation of the dam has been modeled using two-dimensional infinite elements in both FEM and TDSFEM analysis, which is a novel application in the case of TDSFEM. This article quantifies the computational efficiency of TDSFEM over the conventional FEM by comparing the computation time in both methods and thus emphasizes the applicability of TDSFEM in dynamic analysis, especially in case of complex geometries and large two-dimensional structures (like concrete gravity dam). The results demonstrate the computational efficiency of TDSFEM over FEM when higher order elements are considered. Modal analysis and time history analysis results point that the use of higher order elements of TDSFEM can be a viable alternative to the use of the conventional FEM leading to significant saving in computational time with reasonable accuracy for dynamic analysis of large structures like concrete gravity dams.