Implicit Finite Element Research Articles

Numerical modeling of an offshore shellfish farm exposed to extreme wave conditions

Shellfish cultivation is a sustainable method of providing human food and can help remove large amounts of CO2 from the atmosphere. Over the last two decades, longline-based structures have dominated farming systems. So far, the innovative technologies for open-ocean shellfish farming remain stagnant and need to be developed. As such, this paper preliminarily studies the operation and survivability abilities of an innovative shellfish farm under extreme wave conditions. To that end, an efficient numerical scheme with a robust implicit finite element method is established. First, the numerical modeling of a single module of the shellfish farm is conducted and the numerical results are verified against physical model tests. Then, the numerical modeling is implemented in a full-scale shellfish farm containing nine floating rafts with suspended lantern nets in a 3×3 configuration exposed to extreme wave conditions. Different angles of wave attack and shellfish rafts with and without lantern nets are fully considered, allowing an assessment of the operation and survivability abilities of the shellfish farm under extreme wave conditions in various situations. The results highlight that the angle of wave attack significantly affected the energy absorption of the mooring system. Moreover, non-linear instability such as subharmonics, which existed in the motion dynamics, can be manipulated to avoid resonant motions. This study provides insights into the evaluation of the safety design of a shellfish farm at both operational and survivability levels. The numerical method can also model other advanced offshore marine structures with multi-modules, such as floating bridges, airports, and even floating energy islands.

GTP-24-1666 - Temperature Margin Calculation During Finite Element Creep Simulations (GT2024-121776)

Abstract Creep deformation in turbomachinery applications is a highly non-linear phenomenon where +/- 15°C may halve or double the average rupture life. Even in well-controlled laboratory conditions, this stochastic nature means that identical tests may differ by a factor of two. Analysts using implicit finite element user creep subroutines may not appreciate the uncertainty of their solution, or the design changes required to account for uncertainty in either the boundary conditions or the solution itself. Designing to a maximum strain limit does not consider the sudden acceleration of tertiary creep rates. Applying a time factor may extend the simulation time by multiple factors. This paper presents a methodology where two estimations of creep damage are made in parallel to the creep simulation at actual operating temperatures. These two estimations consider the predicted change in stress relaxation, at higher or lower temperatures, to estimate the temperature change required to reach the onset of creep, at any given node in the model, and at any given time in the analysis. This margin calculation is expanded to consider the uncertainty in creep prediction. A time-based scatter factor is incorporated into the temperature margin calculation to provide a minimum temperature margin that includes model uncertainty. The analyst can also consider the uncertainty of the temperature prediction and boundary conditions to produce a robust creep prediction in a real-world simulation. The methodology is validated through the FEA of example cases and applied to a creep-limited 2nd stage turbine blade.

Application of a reaction-based water quality model to the total dissolved solids concentration of the Pasig River.

With the goal to support effective water resource management, water quality models have gained popularity as tools for evaluating the distributions of pollutants and sediments. This work focuses on the application of the numerical solution of an advection-dispersion-reaction (ADR) water quality model for rivers and streams to a major Philippine waterway, the Pasig River. The water quality constituent is described by a system of reaction and advection-dispersion-reaction equations. The model and method are based on a previously used strategy where Guass-Jordan decomposition is applied to the matrix system and the resulting conservative form of the model is solved numerically using the fully implicit scheme and finite element method. The methodology is demonstrated by a case study in Pasig River involving the concentrations of total dissolved solids (TDS) obtained from the Department of Environment and Natural Resources (DENR) through the Pasig River Unified Monitoring Stations (PRUMS) report. Sensitivity analysis and parameter estimation are also applied to the model to assess which parameters influence the model output the most.

Modeling and Simulation of the Aging Behavior of a Zinc Die Casting Alloy

While zinc die-casting alloy Zamak is widely used in vehicles and machines, its solidified state has yet to be thoroughly investigated experimentally or mathematically modeled. The material behavior is characterized by temperature and rate sensitivity, aging, and long-term influences under external loads. Thus, we model the thermo-mechanical behavior of Zamak in the solid state for a temperature range from −40 °C to 85 °C, and the aging state up to one year. The finite strain thermo-viscoplasticity model is derived from an extensive experimental campaign. This campaign involved tension, compression, and torsion tests at various temperatures and aging states. Furthermore, the thermo-physical properties of temperature- and aging-dependent heat capacity and heat conductivity are considered. One significant challenge is related to the multiplicative decompositions of the deformation gradient, which affects strain and stress measures relative to different intermediate configurations. The entire model is implemented into an implicit finite element program and validation examples at more complex parts are provided so that the predicability for complex parts is available, which has not been possible so far. Validation experiments using digital image correlation confirm the accuracy of the thermo-mechanically consistent constitutive equations for complex geometrical shapes. Moroever, validation measures are introduced and applied for a complex geometrical shape of a zinc die casting specimen. This provides a measure of the deformation state for complex components under real operating conditions.

On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

AbstractWe consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the and the norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant‐Friedrichs‐Lewy (CFL) condition between the space mesh size and the time step sizes.

An explicit finite element discrete crack analysis of open hole tension failure in composites

Developing efficient, robust methods for predicting the progressive failure of carbon fiber composites under tension is essential for optimal design, balancing weight reduction, damage tolerance, and service life. Implicit finite element (FE) methods struggle with the complex interplay of failure mechanisms, leading to a heavy reliance on costly, time-consuming experimental testing. This study introduces a novel explicit algorithm designed for laminate-level mesh models, capable of simulating cohesive cracks without predefined paths. Utilizing an explicit finite element software approach, our method effectively addresses convergence issues associated with the non-linear and unstable nature of composite failures during quasi-static loading. Contrasting with implicit FE method, this mesh-independent approach is both practical and numerically robust. The algorithm’s performance, tested on carbon-fiber reinforced composites with varying ply configurations [452/-452]s, [45n/90n/-45n/0n]s and [45/90/-45/0]ns (n=1/2/4/8), demonstrates improved simulation accuracy and efficiency over implicit FE methods, aligning well with prior experimental and simulation data. This advancement offers a promising solution for accurately simulating and understanding the complex failure behavior of these composites.

Ejector pin-related high-cycle fatigue fracture of the critical die corner during automatic multistage cold forging of an automotive wheel nut

An experimental and numerical study on the ejector pin's mechanics during automatic multistage cold forging (AMSCF) of an automobile wheel nut is conducted. The traditional, decoupled die structural analysis method (DDSM) of analyzing die structures as one of the post-processing functions is criticized, which uses the tractions exerting on the die parts predicted from the forging simulation under the rigid die assumption. To cope with the matter of the DDSM, a multibody treatment scheme (MBTS) is proposed to simulate the AMSCF process, emphasizing the ejector pin's mechanics, using an implicit elastoplastic finite element method. The experiments qualitatively validate the finite element predictions. It is shown that the asymmetric sheared material in AMSCF greatly influences the ejector pin's mechanics, which is characterized by its lateral and longitudinal displacements because of its structural flexibility. It is emphasized that the detailed understanding of the ejector pin's mechanics may not only give a helpful connection towards smart manufacturing because of its mechanical flexibility and sensitivity to the excitations and responses, but it also reveals the reason for the die's high-cycle fatigue (HCF) fracture of the critical die corner (CDC) near at the end of the ejector pin.

Computational analysis of the failure mechanisms of a laminated composite in double-edge notch compression configuration

This manuscript presents a combined experimental–computational investigation of the compression failure response of laminated composites in double-edge notch compression (DENC) configuration. Analysis of in-situ and post-test DENC experiment results indicates the presence of multiple failure mechanisms, including ply splitting, delamination, and compression kink bands. In order to characterize the onset, growth, and interplay of the critical and subcritical damage mechanisms, a multiscale progressive damage analysis approach is implemented. A nonlocal multiscale damage model explicitly tracks the intraply failure mechanisms that lead to the formation and propagation of kink bands in the specimen. A cohesive zone model is used to track initiation and progression of ply splitting in the specimen. The splitting model was implemented using implicit finite element simulations and deployed with a pre-crack insertion technique to reduce simulation time while preserving prediction accuracy. Model parameters are calibrated based on available calibration data and measurements. The effects of split location and growth characteristics on model predictions are studied. The computational model along with a suite of experiments were employed to study the formation, growth and interactions of damage mechanisms in the composite specimens subjected to compression loading. The onset of matrix damage is not clear from the experimental images but the model predicts early onset around the onset time of splits which greatly reduce the stress concentration at the notches. Analysis of acoustic emission energy experimental data indicate that observable compression failure mechanisms are not causing nonlinearity consistently exhibited in the experiment responses. Simulation results demonstrate the capability of the multiscale modeling approach to predict critical kink bands and sub-critical matrix cracking and splitting in the DENC specimens while reducing computational cost compared to a direct numerical simulation of fibers and matrix.

Geom-DeepONet: A point-cloud-based deep operator network for field predictions on 3D parameterized geometries

Modern digital engineering design process commonly involves expensive repeated simulations on varying three-dimensional (3D) geometries. The efficient prediction capability of neural networks (NNs) makes them a suitable surrogate to provide design insights. Nevertheless, few available NNs can handle solution prediction on varying 3D shapes. We present a novel deep operator network (DeepONet) variant called Geom-DeepONet, which encodes parameterized 3D geometries and predicts full-field solutions on an arbitrary number of nodes. To the best of the authors’ knowledge, this is the first attempt in the literature and is our primary novelty. In addition to expressing shapes using mesh coordinates, the signed distance function for each node is evaluated and used to augment the inputs to the trunk network of the Geom-DeepONet, thereby capturing both explicit and implicit representations of the 3D shapes. The powerful geometric encoding capability of a sinusoidal representation network (SIREN) is also exploited by replacing the classical feedforward neural networks in the trunk with SIREN. Additional data fusion between the branch and trunk networks is introduced by an element-wise product. A numerical benchmark was conducted to compare Geom-DeepONet to PointNet and vanilla DeepONet, where results show that our architecture trains fast with a small memory footprint and yields the most accurate results among the three. Results show a much lower generalization error of our architecture on unseen dissimilar designs than vanilla DeepONet. Training the model on 2500 cuboid designs is done within 2 h on a single A100 GPU while generating the training dataset through finite element simulations took about 27 h. Once trained, the model can predict vector fields and a single prediction can be over 105 times faster than its corresponding implicit finite element simulation if the mesh is large. The ability of the proposed model to perform efficient and accurate field predictions on variable 3D geometries, especially those discretized by different nodes and elements, makes it a valuable tool for preliminary performance evaluation and design optimizations and is the most significant contribution of the current work.

An implicit solution method for digital construction of temperature fields in concrete structures based on point temperature measurements

PurposeAt present, using the finite element method is difficult to efficiently and accurately construct the temperature field of mass concrete based on temperature measurement points. Thus, there is a need to propose a method for improvement.Design/methodology/approachThis study developed an implicit finite element method that digitally constructs the temperature field of mass concrete based on temperature measurement data. That is, in the proposed method of this paper, the temperature of the measuring point is also one of the boundary conditions, which real-time corrects the calculation error.FindingsIn this method, during the digital construction of the temperature field, the computed temperature approaches the actual measured value at the point of measurement with increasing iteration steps. Using this method and sufficient temperature measurement data, the errors in calculation conditions (such as the boundary conditions, the initial casting temperature and material parameters) can be automatically corrected during the iterative computation process.Originality/valueThis new method can improve calculation accuracy and allows the digitally constructed temperature field to converge to its true value with sufficient measurement data.

A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients

Abstract The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the stochastic convolution. This improves relevant results in the literature and covers both the space-time white noise ($d=1$) and the trace-class noises ($\text{Tr} (Q) < \infty $) in multiple dimensions $d=2,3$. Such an improvement is achieved based on a key perturbation estimate for a perturbed PDE, with the aid of which we prove the convergence and uniform regularity of a spectral approximation of the SPDEs and thus get the improved regularity results. The second contribution of the paper is to propose and analyze a spatio-temporal discretization of the SPDEs, by incorporating a standard finite element method in space and a linearly implicit nonlinearity-tamed Euler method for the temporal discretization. The proposed time-stepping scheme is linearly implicit and does not suffer from solving nonlinear algebra equations as the backward Euler scheme does. Based on the improved regularity results, we recover the expected strong convergence rates of the fully discrete scheme and reveal how the convergence rates rely on the regularity of the noise process. In particular, a classical convergence rate of order $O(h^{2} +\tau )$ can be obtained even in high dimension $d=3$, as the driven noise is of trace class and satisfies certain regularity assumptions. The optimal error estimates turn out to be challenging and face some essential difficulties when the tamed time-stepping scheme meets the finite element spatial discretization, particularly in the context of low regularity and multiple dimensions $d \le 3$. Some highly nontrivial arguments are introduced to overcome the difficulties. Finally, numerical examples corroborate the claimed strong orders of convergence.

Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers

Generalized continuum models for representing nonlinear material behavior including material failure in the finite strain regime are commonly formulated based on scalar elastic and dissipation potential functions. The evolution of stresses and internal variables, i.e., the material state, is governed by partial derivatives of the potential functions with respect to deformation and stress measures. Furthermore, for application of such models in implicit nonlinear finite element analysis tangent operators, consistent with the numerical integration algorithm, are required. In this work, we study analytical, numerical and automatic differentiation schemes for the implementation of coupled, generalized continuum models considering finite inelastic deformations and fracture. This includes the application of finite difference approximations, complex-step derivative approximations and automatic differentiation based on hyper-dual numbers. For the use of automatic differentiation, a semi-analytical split approach is introduced for increasing the computational efficiency. Based on a comprehensive 2D and 3D finite element study, we demonstrate the superior properties of automatic differentiation compared to numerical differentiation methods with regards to accuracy and robustness. Furthermore, the additional computational cost for automatic differentiation using the proposed split approach becomes negligible for large problems.

Simulation of crack closure effect on cracked beam vibrations

Vibrations of beams containing cracks are usually investigated by using open or breathing crack models. Neither one of these approaches considers the plasticity induced crack closure (PICC) phenomena which occurs in fatigue crack growth. In this study, PICC is integrated into the vibration analysis of a cracked beam. First, fatigue crack growth at a notch is simulated with elastoplastic finite element analysis. Residual deformations, stresses and the contact pressures on crack faces are obtained. In the second phase, the model which includes these quantities is used for forced vibration analysis under harmonic loading by using implicit finite element method. For comparison, analyses are also running without PICC. Simulation results indicate that PICC has a significant effect on vibrations of cracked beams. For low load amplitude levels, cracked beam behaves like an intact beam. Nonlinear behavior appears only at rather high amplitude levels. It is concluded that PICC could suppress the crack breathing behavior and this could make it difficult to detect a fatigue crack in a beam from its nonlinear vibration response.

Implicit multiscale finite element analysis of polymer physics-based multiscale constitutive model for elastomers

This paper proposes a multiscale visco-hyperelastic constitutive model that can predict the material behavior bridging molecular properties to response at the continuum level and its integration within implicit finite element analysis. The proposed model incorporates all parameters having actual physical meaning obtained from molecular simulations based on the tube theory. Despite the advantages that the constitutive equation was determined from molecular network characteristics, applications of the model have been restricted to the simple shape of Finite Element (FE) models with a few cases of specific deformations, owing to its complexity and absence of the implicit formulation. Therefore, the application of the model to the 3D implicit FE analysis with a tangent stiffness was proposed by applying numerical differentiation with the complex step derivative approximation (CSDA) method. Combined with the Neo-Hookean hyperelasticity model, the results from the example analysis applied to the tensile specimen model were examined to manifest the visco-hyperelastic characteristics of elastomers under cyclic loading, stability of the analysis, and further discussion on the visco-hyperelastic constitutive model were proposed.

A Conservative Finite Element Scheme for the Kirchhoff Equation

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

An implicit stabilized node-based smoothed finite element method for ultimate bearing capacity analysis of strip footing

The node-based smoothed finite element method (NS-FEM) has gained significant attention over the past decade. Nodal integration in NS-FEM, however, may lead to spurious oscillation of deformations of soil mass. To address the spurious oscillation of deformations associated with nodal integration, some stabilization techniques have been developed, but most of the stabilized NS-FEM have been implemented in the explicit finite element framework. The implicit framework, nonetheless, may offer some advantages including the high efficiency, good numerical accuracy, and suitability for handling large load increments and static problems. In this study, a novel stabilized NS-FEM by referring to the Taylor series expansion and incorporating the secant shear modulus, named sNS-FEM-KF(Gs), is developed in the implicit finite element framework. Based on the ultimate bearing capacity analysis of some strip footings, it is found that sNS-FEM-KF(Gs) may effectively eliminate the spurious oscillations observed in NS-FEM, and may resolve the overly-stiff behaviors of soil mass predicted by sNS-FEM-KF which is a reduced version of sNS-FEM-KF(Gs). Even compared to the standard finite element method with the 6-node triangular (T6) elements (i.e. FEM(T6)), sNS-FEM-KF(Gs) with linear 3-node triangular (T3) elements may outperform FEM(T6) in the investigated strip footing problems, indicating its good applicability in geotechnical analysis.

Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model

In the present work, we investigate a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We show that when the convective part of the system, the haptotaxis term, is dominant, then straightforward numerical methods for the studied system may be unstable. We present an implicit finite element method using conforming P1 or Q1 finite elements to discretize the model in space and the θ-method for discretization in time. The discrete problem is stabilized using a nonlinear flux-corrected transport approach. It is proved that both the nonlinear scheme and the linearized problems used in fixed-point iterations are solvable and positivity preserving. Several numerical experiments are presented in 2D to demonstrate the performance of the proposed method.

Numerical investigation of monopiles in structured clay under cyclic loading

Offshore wind is undergoing rapid growth and is expected to play a vital role in the transition to a low-carbon global economy. This paper presents the results of implicit three-dimensional finite element analyses investigating cyclic lateral loading on a large diameter monopile in structured London Clay. The cyclic behaviour of the clay was simulated using the subloading tij constitutive model. Symmetric and asymmetric two-way and one-way low frequency cyclic lateral loads were used with an amplitude representing offshore wind turbine production loading conditions. Simulations showed that one-way loads induced greater cumulative pile deformation and rotation relative to two-way loads. For all load configurations, the rate of accumulation of lateral pile displacement and rotation at mudline reduced with each subsequent cycle. In contrast, the cyclic secant stiffness progressively degraded with its rate of decay reducing with each ensuing cycle. Findings are indicative of the incidence of strain softening, reconsolidation and densification during low-frequency cyclic loading. It was established that assuming the clay to be unstructured would be conservative in the design of monopiles.

Multiphysics fuel performance modeling of dispersed TRISO-coated particle fuel plate under long-time normal condition and accident conditions

TRISO-coated nuclear fuel has been extensively studied these decades due to the reliable accident tolerance performance. Based on the ABAQUS implicit finite element solver, the thermal–mechanical coupled simulations of the TRISO-particles embedded plate fuel were carried out. Multi-physical effects were incorporated into the simulation, including the plastic deformation, irradiation swelling, creep and gap heat transfer. Results show that the dispersed TRISO-coated particle fuel plate has robust thermal–mechanical performance under long-time normal condition. The maximum fuel temperature is 668 K and the cumulative failure probability for the SiC layer is 1.85 × 10−4. Under reactivity insertion accident and loss of coolant accident conditions, this fuel has reliable anti-accident capacities due to TRISO layer’s protection. Additionally, the porosity effect on thermomechanical performance of this fuel was analyzed.

Performance of LG window systems with different materials under extreme static loading

Laminated glass (LG) is created by bonding two or more glass layers together with one or more polymer interlayer(s). LG may significantly lower the risk of injury from broken glass because the polymeric interlayer retains the glass pieces, especially when subjected to explosive threats. Many interlayer materials have recently been used in LG panels, but the effects on the performance of glass thickness and type, interlayer thickness and type, and hybrid interlayers are not well known. In this work, we report on new implicit and explicit nonlinear finite element models for simulating pressurized water-chamber-like quasi-static tests. The numerical models were validated using 18 by 18 in. LG tests. This study shows that the resistance function of LG can be well predicted numerically and can be used as a tool to study the effects of various glass and interlayer types and configurations. Explicit solutions had a slightly greater resolution in predicting the fracture behavior and post-peak response and captured the distinct failure points of the inner and outer glass panes. Implicit calculations maintained high levels of accuracy and were orders of magnitude faster than the explicit simulations. Using the implicit method, a parametric study was conducted on 38 by 66 in. LG to determine the effects of glass and interlayer thickness, size, and layup configuration. The LG resistance function was improved within the pre-cracked stage by increasing interlayer thickness. Ionoplast SentryGlas® interlayers were shown to be the strongest of the three types considered. Additionally, unique hybrid interlayers were simulated and indicated that strength and bonding can be maximized using various hybrid configurations.