Implicit Finite Element Method Research Articles

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.

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.

Nonlinear dynamics of an aquaculture cage array under oblique wave attack

Nonlinear interactions of an oblique wave with an aquaculture cage array containing 16 net cages in a 2 × 8 configuration is studied in this paper. Our objective is to understand the nonlinear dynamics of the cage array induced by different hydrodynamic forcing mechanisms under oblique wave attack. To this end, we use an efficient numerical scheme with a robust implicit finite-element method for the nonlinear wave-structure interactions. The nonlinear and non-stationary data is decomposed by the empirical mode decomposition method, allowing for identification of different hydrodynamic forcing mechanisms. Then, physical model tests of a cage array in the 1 × 3 configuration under the low-frequency wave are conducted to verify the present numerical model. The harmonic responses of the surge and heave between numerical simulations and experimental tests are close to each other; indicating that the present numerical model is feasible to predict the dynamic response of the cage array in waves. The results highlight that the flexible cage with lower elastic modulus shows larger excursion in the weather side of the cage array, and the flexible cage has a better ability to absorb energy from incident waves. However, the elastic modulus of the floating collar has little effect on the surge and heave motions. Under the oblique wave attack, the nonlinear responses of a representative mooring are identified to be wave-frequency and low-frequency driven, and the nonlinear sway responses are found to be dominant by low-frequency excitations.

A particle finite element method based partitioned paradigm for the axial-flow-induced vibration analysis of NHR200-II fuel rod

As a newly designed 200 MW nuclear heating reactor (NHR200-II), flow-induced vibration (FIV) of the fuel rod has attracted extensive attention due to its slender shape and the hydrodynamic loads arose from the turbulent flow of the surrounding fluid. Fretting wear and/or damage of fuel rod induced by FIV would highly affect system operation and nuclear safety. In this article, a particle finite element method (PFEM) based partitioned paradigm (i.e., implicit finite element method for structure dynamics, PFEM for fluid flow, and unsteady Reynolds averaged Navier–Stokes for turbulence modelling) toward FIV problems was proposed, implemented, and validated. Axial FIV of a single NHR200-II fuel rod was then analyzed through this finite-element based framework. Vibration characteristics of the fuel rod against varied turbulent inflow velocities with a constant turbulence intensity Tv of 5% were discussed in detail. The results showed that horizonal displacement is larger than vertical displacement but both within the same order of magnitude. The effect of inflow velocity of 1.0–2.0 m/s on the dominant frequency is also captured. Besides, fluctuating horizonal pressure is identified as the main source of forced vibration. Therefore, reinforcements on the horizontal constraint are recommended to better eliminate the vibration and enhance the reactor safety.

Carbon Nano-Onions as Nanofillers for Enhancing the Damping Capacity of Titanium and Fiber-Reinforced Titanium: A Numerical Investigation

Improving the damping capacity of metal matrix composites is crucial, especially for applications in the aerospace industry where reliable performance against vibrations and shocks is mandatory. The main objective of the present study is the numerical prediction of the damping behavior of alpha titanium matrix nanocomposites reinforced with hollow carbon nano-onions at various volume fractions. According to the proposed numerical scheme, a structural transient analysis is implemented using the implicit finite element method (FEM). The metal matrix nanocomposites are modeled via the utilization of appropriate representative volume elements. To estimate the mechanical and damping behavior of the nanocomposite representative volume elements, axial sinusoidally time-varying loads are applied to them. The damping capacity of the metal matrix nanocomposites is then estimated by the arisen loss factor, or equivalently the tan delta, which is computed by the time delay between the input stress and output strain. The analysis shows that the loss factor of alpha titanium may be improved up to 60% at 100 Hz by adding 5 wt% carbon nano-onions. The numerical outcome regarding the dynamic properties of the carbon nano-onions/alpha titanium nanocomposites is used in a second-level analysis to numerically predict their damping performance when they are additionally reinforced with unidirectional carbon fibers, using corresponding representative volume elements and time-varying loadings along the effective direction. Good agreement between the proposed computational and other experimental predictions are observed regarding the stiffness behavior of the investigated metal matrix nanocomposites with respect to the mass fraction of the carbon-onion nanofillers in the titanium matrix.

A fully coupled THMC model for simulating hydrate dissociation by using finite element method

PurposeNatural gas hydrate (NGH) has been regarded as one of the most important resources due to NGH's large amounts of reserve. However, NGH development still faces many technical challenges, such as low production rate and reservoir instability resulting from NGH decomposition. Therefore, developing a fully coupled THMC model for simulating the hydrate decomposition and studying its mechanical behavior is very important and necessary. The purpose of this article is to develop and solve a multi-phase, strong nonlinearity and large-scale fully coupled thermal-hydro-mechanical–chemical (THMC) model for simulating the multi-physics processes involving solid-liquid-gas flow, heat transfer, NGH phase change and rock deformation during NGH decomposition.Design/methodology/approachIn this paper, a multi-phase, strong nonlinearity and large-scale fully coupled THMC model is developed for simulating the multi-physics processes involving solid-liquid-gas flow, heat transfer, NGH phase change and rock deformation during NGH dissociation. The fully coupled THMC model is solved by using a fully implicit finite element method, in which the gas pressure, water pressure, temperature and displacement are taken as basic unknown variables. The proposed model is validated against with the experimental data, showing high accuracy and reliability.FindingsA multi-phase, strong nonlinearity and large-scale fully coupled THMC model is developed for simulating the multi-physics processes involving solid-liquid-gas flow, heat transfer, NGH phase change and rock deformation during NGH decomposition. The proposed model is validated against with the experimental data, showing high accuracy and reliability.Research limitations/implicationsSome assumptions are made to make the model tractable, including (1) the composition gas of hydrate is pure methane; (2) the gas-liquid multi-phase flow in the pore obeys Darcy's law; (3) hydrate occurs on the surface of soil particles, both of them form the composite consolidation material; (4) the small-strain assumption is applied to composite solid materials, which are treated as skeletons and cannot be moved; (5) momentum change caused by phase change is not considered.Practical implications NGH has been regarded as one of the most important resources due to its large amounts of reserve. However, NGH development still faces many technical challenges, such as low production rate and reservoir instability resulting from NGH decomposition. Most of the existing studies decouple the process with solid deformation and seepage behavior, but the accuracy of the numerical results will be sacrificed to certain extent. Therefore, it is very important and necessary to develop a fully coupled THMC model for simulating the hydrate decomposition and studying its mechanical behavior.Social implications NGH, widely distributed in shallow seabed or permanent frozen region, has the characteristics of high energy density and high combustion efficiency (Yan et al., 2020). A total of around 7.5 × 1,018 m3 has been proved to exist around the world and 1 m3 of NGH can release about 160–180 m3 of natural gas (Kvenvolden and Lorenson) under normal conditions. Safely and sustainably extracting NGH commercially can effectively relieve global energy pressure and contribute to achieving carbon reduction goals.Originality/valueThe novelty of the present work lies in mainly two aspects. First, a fully coupled THMC model is developed for studying the multi-physics processes involving solid-liquid-gas flow, heat transfer, NGH phase change and solid deformation during NGH dissociation. Second, the numerical solution is obtained by using a fully implicit finite element method (FEM) and is validated against experimental data.

Fracture of porcine aorta—Part 2: FEM modelling and inverse parameter identification

The mechanics of vascular tissue, particularly its fracture properties, are crucial in the onset and progression of vascular diseases. Vascular tissue properties are complex, and the identification of fracture mechanical properties relies on robust and efficient numerical tools. In this study, we propose a parameter identification pipeline to extract tissue properties from force-displacement and digital image correlation (DIC) data. The data has been acquired by symconCT testing porcine aorta wall specimens. Vascular tissue is modelled as a non-linear viscoelastic isotropic solid, and an isotropic cohesive zone model describes tissue fracture. The model closely replicated the experimental observations and identified the fracture energies of 1.57±0.82 kJ m−2 and 0.96±0.34 kJ m−2 for rupturing the porcine aortic media along the circumferential and axial directions, respectively. The identified strength was always below 350 kPa, a value significantly lower than identified through classical protocols, such as simple tension, and sheds new light on the resilience of the aorta. Further refinements to the model, such as considering rate effects in the fracture process zone and tissue anisotropy, could have improved the simulation results. Statement of significanceThis paper identified porcine aorta’s biomechanical properties using data acquired through a previously developed experimental protocol, the symmetry-constraint compact tension test. An implicit finite element method model mimicked the test, and a two-step approach identified the material’s elastic and fracture properties directly from force-displacement curves and digital image correlation-based strain measurements. Our findings show a lower strength of the abdominal aorta as compared to the literature, which may have significant implications for the clinical evaluation of the risk of aortic rupture.

Seismic damage characteristics of high arch dams under oblique incidence of SV waves

This study investigates the seismic damage characteristics of high arch dams considering the oblique incidence of seismic waves. Based on the dynamic implicit finite element method, a viscous-spring artificial boundary is applied to consider radiation damping of the infinite foundation, the seismic wave input is converted into an equivalent nodal force of the artificial boundary, and the equivalent nodal loads of the wave input under oblique incidence of the SV wave is deduced. Taking JP Ⅰ arch dam as an example, combined with the concrete damaged plasticity (CDP) model, the nonlinear dynamic response of the dam under 25° oblique incidence of the SV wave is conducted considering the geometrical nonlinearity of contraction joints and the dynamic mechanical properties of concrete materials. Based on the incremental dynamic analysis (IDA) method, the relative displacement and transverse joint opening are selected as the damage measures for fragility analysis of the arch dam. According to the damage distribution from the analysis of a large number of seismic time-history responses, the damage levels of the arch dam are classified. Three limit states of slight damage, moderate damage and severe damage are proposed for the arch dam. The average responses of the 10 earthquakes are considered as the limit values to quantitatively describe the damage of the arch dam. By fitting the results of incremental dynamic analysis, the seismic probabilistic demand model and seismic fragility curves for the two damage measures to predict the probability of the arch dam reaching damage levels under different seismic actions have been established. The fragility curve shows that the probability of severe damage is almost zero under the maximum design earthquake (PGA = 0.1971 g) and less than 5% under the maximum credible earthquake (PGA = 0.2299 g). It is concluded that JP Ⅰ arch dam has a high degree of safety redundancy and fully satisfies the “two levels” design requirements.

Realistic finite element analysis model of the pilgering process to deal with initial tube thickness nonuniformity

Traditional finite element analysis (FEA) models for simulating the pilgering process, based on the assumption of a fixed rigid mandrel and uniformity of the initial tube thickness, have practical disadvantages in solving industrial problems due to tube thickness nonuniformity and related mandrel deformation. A novel FEA model of the pilgering process with initial tube nonuniformity is presented herein, using an implicit elasto-thermoviscoplastic finite element method (FEM) with tetrahedral MINI-elements and multi-body treatment scheme, with an emphasis on the moveable and deformable mandrel and practical boundary conditions for the tube material. The effects of the number of layers in the radial direction on the predictions are investigated. The variation in thickness ratio with stroke during the pilgering process is predicted. It has been shown that the pilgering process considerably improves tube eccentricity.

Patient-specific finite element simulation of peripheral artery percutaneous transluminal angioplasty to evaluate the procedure outcome without stent implantation.

The purpose of this work is to present a patient-specific (PS) modeling approach for simulating percutaneous transluminal angioplasty (PTA) endovascular treatment and assessing the balloon sizing influence on short-term outcomes in peripheral arteries, i.e. without stent implantation. Two 3D PS stenosed femoral artery models, one with a dominant calcified atherosclerosis while the other with a lipidic plaque, were generated from pre-operative computed tomography angiography images. Elastoplastic constitutive laws were implemented within the plaque and artery models. Implicit finite element method (FEM) was used to simulate the balloon inflation and deflation for different sizings. Besides vessel strains, results were mainly evaluated in terms of the elastic recoil ratio (ERR) and lumen gain ratio (LGR) attained immediately after PTA. Higher LGR values were shown within the stenosed region of the lipidic patient. Simulated results also showed a direct and quantified correlation between balloon sizing and LGR and ERR for both patients after PTA, with a more significant influence on the lumen gain. The max principal strain values in the outer arterial wall increased at higher balloon sizes during inflation as well, with higher rates of increase when the plaque was calcified. Results show that our model could serve in finding a compromise for each stenosis type: maximizing the achieved lumen gain after PTA, but at the same time without damaging the arterial tissue. The proposed methodology can serve as a step toward a clinical decision support system to improve angioplasty balloon sizing selection prior to the surgery.

A two-grid ADI finite element approximation for a nonlinear distributed-order fractional sub-diffusion equation

<abstract><p>In this paper, a two-grid alternating direction implicit (ADI) finite element (FE) method based on the weighted and shifted Grünwald difference (WSGD) operator is proposed for solving a two-dimensional nonlinear time distributed-order fractional sub-diffusion equation. The stability and optimal error estimates with second-order convergence rate in spatial direction are obtained. The storage space can be reduced and computing efficiency can be improved in this method. Two numerical examples are provided to verify the theoretical results.</p></abstract>

Martian buildings: structural forms using in-place sources

On Mars, structural loads and the low tensile strength of in-place Martian binders make existing solutions for Martian structures uneconomical because they are based on the terrestrial sources like inflatable units. Here we address this issue by introducing and analyzing three innovative structural forms in accordance with the structural engineering point of view using symmetric optimum parabolic rotated arch shapes and in-place waterless sulfur-based concrete. These forms minimize the tensile stresses under Martian structural loads. Probable Martian structural loads, including gravity, wind, marsquakes, asteroid and meteoroid impact loads and their effects have been investigated and calculated. The proposed models were analyzed under Martian structural loads using the implicit finite element method and the results were compared to two concrete structural forms from previous studies. The proposed models could tolerate Martian structural loads with complete elastic behavior and would significantly decrease the Martian colonization cost due to using Martian resources and reduce element importing from Earth.

Fine-grained heterogeneous parallel direct solver for finite element problems

The improvement of computational effectiveness is a vital issue in the field of large-scale finite element analysis. The performance is fundamentally determined by the efficiency of solving sparse linear system equations using the implicit finite element method. This paper presents a direct linear solver based on heterogeneous hybrid parallel computing on CPUs and GPUs. This can efficiently utilize computing resources of multiple devices to achieve performance improvement. Initially, we partition the elimination tree into several subtrees to accomplish the task decomposition. Based on this, we build a dynamic programming mathematical model to balance the computational load of the various devices. Then, we develop a numerical decomposition strategy by combining node parallelism and tree parallelism for the CPUs. In addition, efficient numerical decomposition is achieved on the GPU through batch processing and maximizing the overlap between computations and data transfers. Numerical experiments show that, compared with MKL PARDISO, the performance of numerical factorization can be improved by up to 10 times by using CPU and dual-path GPU hybrid calculations, and the computation time of simulation can be reduced by one-third for the multicondition analysis of Body In White and by 20% for the large-scale nonlinear finite element deformation analysis.

A two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential

AbstractIn this paper, we present a two‐grid finite element method for the Allen‐Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen‐Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two‐grid finite element method. The optimal convergence order of the two‐grid finite element method in H1 norm is achieved when the mesh sizes satisfy h = O(H2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two‐grid method can save the CPU time while keeping the same convergence rate.

High-resolution cerebral blood flow simulation with a domain decomposition method and verified by the TCD measurement

Background: An efficient and accurate blood flow simulation can be useful for understanding many vascular diseases. Accurately resolving the blood flow velocity based on patient-specific geometries and model parameters is still a major challenge because of complex geomerty and turbulence issues. In addition, obtaining results in a short amount of computing time is important so that the simulation can be used in the clinical environment. In this work, we present a parallel scalable method for the patient-specific blood flow simulation with focuses on its parallel performance study and clinical verification.Methods: We adopt a fully implicit unstructured finite element method for a patient-specific simulation of blood flow in a full precerebral artery. The 3D artery is constructed from MRI images, and a parallel Newton–Krylov method preconditioned with a two-level domain decomposition method is adopted to solve the large nonlinear system discretized from the time-dependent 3D Navier-Stokes equations in the artery with an integral outlet boundary condition. The simulated results are verified using the clinical data measured by transcranial Doppler ultrasound, and the parallel performance of the algorithm is studied on a supercomputer.Results: The simulated velocity matches the clinical measured data well. Other simulated blood flow parameters, such as pressure and wall shear stress, are within reasonable ranges. The results show that the parallel algorithm scales up to 2160 processors with a 49% parallel efficiency for solving a problem with over 20 million unstructured elements on a supercomputer. For a standard cerebral blood flow simulation case with approximately 4 million finite elements, the calculation of one cardiac cycle can be finished within one hour with 1000 processors.Conclusion: The proposed method is able to perform high-resolution 3D blood flow simulations in a patient-specific full precerebral artery within an acceptable time, and the simulated results are comparable with the clinical measured data, which demonstrates its high potential for clinical applications.

Crystal plasticity modeling of strain-induced martensitic transformations to predict strain rate and temperature sensitive behavior of 304 L steels: Applications to tension, compression, torsion, and impact

This paper advances crystallographically-based Olson-Cohen (direct γ → α’) and deformation mechanism (indirect γ→ε→α’) phase transformation models for predicting strain-induced austenite to martensite transformation. The advanced transformation models enable predictions of not only strain-path sensitive, but also of strain-rate and temperature sensitive deformation of polycrystalline stainless steels (SSs). The deformation of constituent grains in SSs is modeled as a combination of anisotropic elasticity, crystallographic slip, and phase transformation, while the hardening is based on the evolution of dislocation density and explicit shifts in phase fractions. Such grain-scale deformation is implemented within the meso‑scale elasto-plastic self-consistent (EPSC) homogenization model, which is coupled with the implicit finite element (FE) method to provide a constitutive response at each FE integration point for solving boundary value problems at the macro-scale. Parameters pertaining to the hardening and transformation models within FE-EPSC are calibrated and validated on a suite of data including flow curves and phase fractions for monotonic compression, tension, and torsion as a function of strain-rate and temperature for wrought and additively manufactured (AM) SS304L. To illustrate the potential and accuracy of the integrated multi-level FE-EPSC simulation framework, geometry, mechanical response, phase fractions, and texture evolution are simulated during gas-gun impact deformation of a cylinder and quasi-static tension of a notched specimen made of AM SS304L. Details of the simulation framework, comparison between experimental and simulation results, and insights from the results are presented and discussed.

Partitioned analysis of soil‐structure interaction for Nuclear Island Buildings

AbstractThe present methods for three‐dimensional time‐domain soil‐structure interaction (SSI) analysis are often uneconomical because they are performed with a single time integration scheme and a single time step, which prohibits their application to large‐scale SSI problems. In this study, a partitioned analysis of SSI (PASSI) is proposed for enhancing SSI computational efficiency. This is accomplished by partitioning the soil‐foundation‐structure system into the soil (foundation) and structure subsystems and implementing the continuity conditions of the displacements and reaction forces at the soil (foundation)‐structure interface in a primal way. A lumped‐mass explicit finite element method and a transmitting artificial boundary are used to model the unbounded soil, the structure is analyzed via the implicit finite element method, and the response of the rigid foundation is calculated through an explicit time integration scheme. The solution is separately advanced over time for each subsystem. Different time steps can be chosen for the explicit and implicit integration schemes, which can greatly improve efficiency. Interaction effects are accounted for by the transmission and synchronization of the coupled state variables. In addition, intrafield and interfield parallel procedures for PASSI are developed, and their theoretical efficiencies are analyzed. A simple example is provided to verify the performance of the partitioned approach with an explicit‐implicit co‐computation and to compare it to the fully explicit approach. Finally, the seismic response analysis of a nuclear power plant is presented to validate the feasibility and efficiency of the intrafield and interfield parallel procedures.

Coupling of a multi-GPU accelerated elasto-visco-plastic fast Fourier transform constitutive model with the implicit finite element method

This paper presents an implementation of the elasto-visco-plastic fast Fourier transform (EVPFFT) crystal plasticity model in the implicit finite element (FE) method of Abaqus standard through a user material (UMAT) subroutine to provide a constitutive relationship between stress and strain at FE integration points. To facilitate the implicit coupling ensuring fast convergence rates, an analytical Jacobian is derived. The constitutive response at every integration point is obtained by the full-field homogenization over an explicit microstructural cell. The implementation is a parallel computing approach involving multi-core central processing units (CPUs) and graphics processing units (GPUs) for computationally efficient simulations of large plastic deformation of metallic components with arbitrary geometry and loading boundary conditions. To this end, the EVPFFT solver takes advantages of GPU acceleration utilizing Nvidia’s high performance computing software development kit (SDK) compiler and compute unified device architecture (CUDA) FFT libraries, while the FE solver leverages the message passing interface (MPI) for parallelism across CPUs. The high-performance hybrid CPU-GPU multi-level framework is referred to as FE-GPU-EVPCUFFT. Simulations of simple compression of Cu and large strain cyclic reversals of dual phase (DP) 590 have been used to benchmark the accuracy of the implementation in predicting the mechanical response and texture evolution. Subsequently, two applications are presented to illustrate the potential and utility of the multi-level simulation strategy: 4-point bending of textured Zr bars, in which the model captures the shape variations as a consequence of texture with respect to the bending plane and another bending of DP1180, in which the model reveals details of spatial micromechanical fields.

Spatiotemporal Evolution of Stress Field during Direct Laser Deposition of Multilayer Thin Wall of Ti-6Al-4V.

The present work seeks to extend the level of understanding of the stress field evolution during direct laser deposition (DLD) of a 3.2 mm thick multilayer wall of Ti-6Al-4V alloy by theoretical and experimental studies. The process conditions were close to the conditions used to produce large-sized structures by the DLD method, resulting in specimens having the same thermal history. A simulation procedure based on the implicit finite element method was developed for the theoretical study of the stress field evolution. The accuracy of the simulation was significantly improved by using experimentally obtained temperature-dependent mechanical properties of the DLD-processed Ti-6Al-4V alloy. The residual stress field in the buildup was experimentally measured by neutron diffraction. The stress-free lattice parameter, which is decisive for the measured stresses, was determined using both a plane stress approach and a force-momentum balance. The influence of the inhomogeneity of the residual stress field on the accuracy of the experimental measurement and the validation of the simulation procedure are analyzed and discussed. Based on the numerical results it was found that the non-uniformity of the through-thickness stress distribution reaches a maximum in the central cross-section, while at the buildup ends the stresses are distributed almost uniformly. The components of the principal stresses are tensile at the buildup ends near the substrate. Furthermore, the calculated equivalent plastic strain reaches 5.9% near the buildup end, where the deposited layers are completed, while the plastic strain is practically equal to the experimentally measured ductility of the DLD-processed alloy, which is 6.2%. The experimentally measured residual stresses obtained by the force-momentum balance and the plane stress approach differ slightly from each other.

Study on implicit finite-element method for steel-arch support

The steel-arch structure in soft-rock tunnels is very small and densely arranged. Too many elements would exist if the solid element is adopted for its simulation, and applying these elements in actual engineering practice is difficult. Therefore, an implicit finite-element method (FEM) to simulate the support effect of steel arches was proposed based on the principle of stiffness superposition. First, an equivalent mechanical model of a steel arch was established based on the principle of stiffness equivalence and deformation-coordination hypothesis. Then, a new concept called auxiliary element was presented. According to this concept, an implicit finite-element model of the equivalent element of the steel arch was developed using the reconstruction technology of elements. The equivalent-stiffness equation of the implicit equivalent element was derived based on the interpolation theory and element-equilibrium equation of FEM, and the equivalent stiffness of the implicit equivalent element was superposed on the stiffness matrix of the auxiliary element to simulate the support effect of the steel arch. Finally, a numerical case was presented to verify the rationality of this method and correctness of the programming. Establishing a solid model of the steel arch using this method was not necessary. Thus, the modeling work was greatly simplified, and adjusting the support scheme was convenient, which provided a new numerical simulation method for the support design of steel arch in soft-rock tunnels.