Stress Interpolation Research Articles

A geometrically nonlinear Hellinger–Reissner shell element for the postbuckling analysis of variable stiffness composite laminate structures

Variable stiffness (VS) composite laminates provide larger freedom to design thin-walled structures than constant stiffness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the postbuckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress field interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures.First, the element performance is investigated by analysing a carbon fibre VS composite laminate plate subjected to compressive stresses. Numerical results obtained with MISS-4C are compared with those obtained with the MISS-4 element, showing good accuracy and a high convergence rate. Subsequently, the structural response of a glass fibre VS composite laminate girder of a short-length bridge is optimised through a multi-objective optimisation that exploits the robustness of the MISS-4C element and the efficiency of the multi-modal Koiter algorithm.

Rational generalized mixed finite element method for 2D linear elastic problems

Combining the ideas of rational displacement finite element and generalized mixed finite element, the rational generalized mixed element with 5 nodes is proposed. The finite formulations for the proposed element are derived on the basis of the generalized mixed variational principle, and the two field variables of displacement and stress can be solved simultaneously. Fundamental solutions of planar linear elasticity are adopted as the displacement interpolation function, while polynomials satisfying the equilibrium equation are adopted as the stress interpolation function. Both the displacement and stress boundary conditions can be introduced directly at same time in the proposed method, which contributes to being a more rational model. The solution technique is utilized to analyze a couple of representative static problems, and the results show the properties of excellent accuracy and good convergence. In addition, it is found to be evident for improvement of the accuracy when the stress boundary conditions are applied.

A layer-wise plasticity-based approach for the analysis of reinforced concrete shell structures using a mixed finite element

The analysis of reinforced concrete shell structures accounting for material nonlinearity is addressed. The structural response is numerically evaluated using a mixed shell finite element and a plasticity-based material behaviour. The finite element is a quadrilateral with four nodes and is based on self-equilibrated assumed stresses. The kinematic fields are interpolated only along the element boundary by polynomials up to the third order. The generalised shell stresses are evaluated through layer-wise integration of the Cauchy pointwise stresses. This allows appropriated three-dimensional elastoplastic constitutive equation to be employed and to include multiple reinforcing layers. The integration of the constitutive laws is performed using a dual decomposition method which preserves the assumed stress interpolation. Results, including civil engineering applications, show that the proposed approach is reliable and accurate in evaluating the static nonlinear equilibrium path. Additionally, the mixed finite element shows a quadratic rate of convergence in the collapse load and low error even for coarse meshes.

A dual decomposition of the closest point projection in incremental elasto‐plasticity using a mixed shell finite element

AbstractMixed assumed stress finite elements (FEs) have shown good advantages over traditional displacement‐based formulations in various contexts. However, their use in incremental elasto‐plasticity is limited by the need for return mapping schemes which preserve the assumed stress interpolation. For elastic‐perfectly plastic materials and small deformation problems, the integration of the constitutive equation furnishes a closest point projection (CPP) involving all the element stress parameters. In this work, a dual decomposition strategy is adopted to split this problem into a series of CPPs at the integration points level and in a nonlinear system of equations over the element, in order to simplify its solution. The strategy is tested with a four nodes mixed shell FE, named MISS‐4, characterized by an equilibrated stress interpolation which improves the accuracy. Two decomposition strategies are tested to express the plastic admissibility either in terms of stress resultants or point‐wise Cauchy stresses. The recovered elasto‐plastic solution preserves all the advantages of MISS‐4, namely it is accurate for coarse meshes in recovering the equilibrium path and evaluating the limit load showing a quadratic rate of convergence, as demonstrated by the numerical results.

A novel hybrid isogeometric element based on two-field Hellinger–Reissner principle to alleviate different types of locking

In the present work, novel hybrid elements are proposed to alleviate the locking anomaly in non-uniform rational B-spline-based isogeometric analysis (IGA) using a two-field Hellinger–Reissner variational principle. The proposed hybrid elements are derived by adopting the independent interpolation schemes for displacement and stress fields. The key highlight of the present study is the choice and evaluation of higher-order terms for the stress interpolation function to provide a locking-free solution. Furthermore, the present study demonstrates the efficacy of the proposed elements with the treatment of several two-dimensional linear-elastic benchmark problems alongside the conventional single-field IGA, Lagrangian-based finite element analysis (FEA), and hybrid FEA formulation. It is shown that the proposed class of hybrid elements performs effectively for analyzing the nearly incompressible problem domains that are severely affected by volumetric locking along with the thin plate and shell problems where the shear locking is dominant. A better coarse mesh accuracy of the proposed method in comparison with the conventional formulation is demonstrated through various numerical examples. Moreover, the formulation is not restricted to the locking-dominated problem domains but can also be implemented to solve the problems of general form without any special treatment. Thus, the proposed method is robust, most efficient, and highly effective against both shear and volumetric locking.

Evaluation of Nearshore QuikSCAT 4.1 and ERA-5 Wind Stress and Wind Stress Curl Fields over Eastern Boundary Currents

Fields of coastal wind stress and wind stress curl in the 10–100 km next to the land control the processes of upwelling and downwelling of nutrients and water properties that are vital to highly productive coastal marine ecosystems. Here we ask the question: Do the present surface wind stress products from a satellite-borne scatterometer (QuikSCAT) and an atmospheric reanalysis model (ERA-5) systematically overestimate the magnitude of wind speed and stress in the 10–50 km next to the coast? We compare QuikSCAT wind speed retrievals to the relatively unused wind speed retrievals from satellite altimeters, which are able to approach closer to the coast than scatterometers without land reflections, due to their smaller radar footprints. Altimeter data on tracks approaching and crossing the coast indicate that the increases in coastal QuikSCAT wind speed values and ERA-5 coastal wind stress values are unrealistic. For analyses of wind speed and stress requiring high accuracy, especially those involving wind stress curl, we suggest considering individual Level 2B scatterometer wind retrievals as suspect at distances of 10 km and less from the coast, along with use of the Poor Coastal Processing flag. We found that similar increases in wind stress values next to the coast in gridded ERA-5 fields are not due to errors in the model physics or wind speeds. They are created during the interpolation of wind stress from the original model grid to a regular rectangular grid. We recommend that researchers who are analyzing wind stress and wind stress curl should calculate wind stress themselves from the gridded ERA-5 vector wind speed fields, rather than using the interpolated model wind stress or curl fields.

Hellinger–Reissner principle based stress–displacement formulation for three-dimensional isogeometric analysis in linear elasticity

In case of Lagrangian finite element formulation, three-dimensional (3-D) stress-based hybrid solid elements have shown excellent coarse mesh accuracy for a wide range of applications. However, to the best of our knowledge, there is no work available towards the development of 3-D stress-based hybrid solid elements for isogeometric analysis (IGA). In this work, we propose stress-based hybrid solid elements to alleviate the issue of locking arising in non-uniform rational B-spline (NURBS)-based IGA. The present work primarily focuses on the linear elasticity, though the formulation can be extended to the non-linear regime. We believe that the excellent coarse mesh accuracy provided by the proposed elements will further enhance the IGA in various applications, especially involving structures with high aspect ratios and nearly incompressible materials. The proposed elements are constructed based on a two-field Hellinger–Reissner variational statement, where stress and displacement fields are interpolated separately. The stress interpolation functions have been derived systemically for various orders of NURBS displacement interpolation functions. Further, we show mathematically that the choice of the stress interpolation functions is free from spurious zero-energy modes. Lastly, the results of numerous 3-D linear-elastic benchmark problems are presented to demonstrate the efficacy and robustness of the proposed elements. The results confirm the superior coarse mesh accuracy for structures with high aspect ratios and almost incompressible materials.

Impact and contact problems of explosives by ‘Mixed’ meshless local Petrov-Galerkin finite volume method

The ‘Mixed’ Meshless Local Petrov-Galerkin Finite Volume method (MLPG5) is developed for solving low-velocity impact problems of plastic bonded explosives (PBX). Due to the ductility and plasticity of PBX, there is large deformation instead of pulverization or fragmentation. In this paper, the impact dynamics equations are solved by the meshless approach (MLPG5), both stress as well as displacements are interpolated, through the moving least squares (MLS) interpolation schemes, which eliminates the expensive process of domain integrals and differentiating the shape function. The collocation method is applied to enforce strain-displacement relationships only at each nodal point, instead of the subdomain. In the present mixed method, the independent interpolation of stress improves accuracies of both the plastic stress and contact stress. The contact force is obtained by penalty function method via iteration scheme. Due to the non-linear constitutive relation, the incremental stress is obtained via radial return schemes through Newton Raphson iteration. Finally, several numerical examples are given to demonstrate the feasibility and the accuracy of the present numerical approach compared with the finite element method. Numerical examples also demonstrate the advantages of the present methods: (i) smaller support sizes can be used; (ii) higher accuracies of stress are obtained.

Topological Design of Multi-Material Compliant Mechanisms with Global Stress Constraints.

This paper presents an approach for the topological design of multi-material compliant mechanisms with global stress constraints. The element stacking method and the separable stress interpolation scheme are applied to calculate the element stiffness and element stress of multi-material structures. The output displacement of multi-material compliant mechanisms is maximized under the constraints of the maximum stress and the structural volume of each material. The modified P-norm method is applied to aggregate the local von Mises stress constraints for all the finite elements to a global stress constraint. The sensitivities are calculated by the adjoint method, and the method of moving asymptotes is utilized to update the optimization problem. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. The appearance of the de facto hinges in the optimal mechanisms can be suppressed effectively by using the topology optimization model with global stress constraints, and the stress constraints for each material can be met.

A corotational mixed flat shell finite element for the efficient geometrically nonlinear analysis of laminated composite structures

AbstractA corotational flat shell element for the geometrically nonlinear analysis of laminated composite structures is presented. The element is obtained from the Hellinger–Reissner variational principle with assumed stress and displacement fields. The stress interpolation is derived from the linear elastic solution for symmetric composite materials. The element is isostatic, namely the stress interpolation is ruled by the minimum number of parameters. Displacement and rotation fields are only assumed along the contour of the element. As such, all the operators are efficiently obtained through analytical contour integration. The geometrical nonlinearity is introduced by means of a corotational formulation. The proposed finite element, named MISS‐4c, proves to be locking free and shows no rank defectiveness. A multimodal Koiter's algorithm is used to obtain the initial postbuckling response. Results show good accuracy and high convergence rate in the geometrically nonlinear analysis of composite shell structures.

Stabilised Variational Multi-scale Finite Element Formulations for Viscoelastic Fluids

The objective of this article is to summarise the work that we have been doing as a group in the context of stabilised finite element formulations for viscoelastic fluid flows. Viscoelastic fluids are complex non-Newtonian fluids, characterised by having an irreducible constitutive equation that needs to be solved coupled with the momentum and continuity equations. The finite element approximation of this kind of fluids presents several numerical difficulties. It inherits obviously the problems associated with the approximation of the incompressible Navier–Stokes equations. But, on top of that, now the constitutive equation is highly non-linear, with an advective term that may lead to both global and local oscillations in the numerical approximation. Moreover, even in the case of smooth solutions, it is necessary to meet some additional compatibility conditions between the velocity and the stress interpolation in order to ensure control over velocity gradients. The stabilised methods detailed in this work allow one to use equal order or even arbitrary interpolation for the problem unknowns ( $${\varvec{\sigma }}$$ - $${\varvec{u}}$$ -p) (elastic deviatoric stress-velocity-pressure) and to stabilise dominant convective terms, and all of them can be framed in the context of variational multi-scale methods. Some additional numerical ingredients that are introduced in this article are the treatment of the non-linearities associated with the problem and the possibility to introduce a discontinuity-capturing technique to prevent local oscillations. Concerning the constitutive equation, both the standard as the logarithmic conformation reformulation are discussed for stationary and time-dependent problems, and different versions of stabilised finite element formulations are presented in both cases.

A computational framework for coating fatigue analysis of wind turbine blades due to rain erosion

The rain-induced fatigue damage in the wind turbine blade coating has attracted increasing attention owing to significant repair and maintenance costs. The present paper develops an improved computational framework for analyzing the wind turbine blade coating fatigue induced by rain erosion. The paper first presents an extended stochastic rain field simulation model that considers different raindrop shapes (spherical, flat, and spindle), raindrop sizes, impact angles, and impact speeds. The influence of these raindrop characteristics on the impact stress of the blade coating is investigated by a smoothed particle hydrodynamics approach. To address the expensive computational time, a stress interpolation method is proposed to calculate the impact stress of all raindrops in a random rain event. Furthermore, coating fatigue analysis is performed by including the fatigue crack initiation in the incubation period and the fatigue crack propagation in the mass-loss-rate increasing period due to raindrop impact. Finally, the proposed computational framework is verified by comparing the estimated fatigue life with those obtained in literature. The results from the study show that by incorporating the statistics of rainfall data, the proposed framework could be used to calculate the expected fatigue life of the blade coating due to rain erosion.

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A reduced model for nonlinear analysis and design of thin-walled structures prone to multi-modal buckling

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

Numerical analysis of a monotube mixed mode magnetorheological damper by using a new rheological approach in CFD

In this study, the analytical Herschel-Bulkley fluid model of a monotube mixed mode MR damper was examined. The MR damper was modeled and simulated by using computational fluid dynamics (CFD) and magnetic finite elements analysis (FEA). The magnetic effects were modeled in a coupled manner with CFD. The actual rheological data was used in the CFD solver to find the cell-based viscosity by using a shear stress interpolation method. The MR damper was manufactured, and experiments were conducted to find the force-displacement and force-velocity correlations, and a good agreement was found between the numerical results and experimental data. The experimental results were also compared with analytical and numerical models under various current values. The CFD results are valuable in predicting the actual characteristics of the non-Newtonian flow inside an MR damper, and it can be used for various non-Newtonian fluid CFD models.

Stress constrained multi-material topology optimization with the ordered SIMP method

To date, both the stress-related and multi-material topology optimization problems have been extensively studied. However, there are few works addressing the stress-constrained multi-material topology optimization (SMMTO) problem. Hence, a novel solution to the SMMTO problem is proposed in this paper based on the ordered SIMP method. To be specific, description of the multi-material elastic model is achieved with the ordered SIMP interpolation. More importantly, a novel ordered SIMP-like interpolation function is proposed to realize the relaxed and scaled stress interpolation, so that only a unique set of density variables are required for the SMMTO problem. Scaling of the stress measure plays a vital role in properly addressing the different yield limits. The global stress constraint is established with the p-norm function combined with the stability transformation method to close the gap between the p-norm stress and the exact maximum local stress. The adjoint sensitivity analysis is performed and the optimization problem is solved using a MMA optimizer. Several benchmark numerical examples are investigated and discussed to illustrate the effectiveness of the proposed approach.

A fundamental class of stress elements in lower bound limit analysis

There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

Alternative method to determine the uniaxial-strain compressibility of cohesive sandstone

Compaction and subsidence risk related to hydrocarbon production are frequently assessed using Geertsma models. The underlying theory assumes linear poroelastic reservoir rock to compress by pore pressure depletion under uniaxial-strain conditions. Adequate mechanical testing techniques have been developed to measure the uniaxial-strain compressibility, although the execution of these methods is technically difficult and requires a significant level of experience. In this paper, a simplified technique is proposed to approximate the compressibility from triaxial strain data, which is different from earlier hydrostatic stress/uniaxial-strain conversion approaches. The method is applied to Berea outcrop sandstone and Groningen reservoir material, employing pore pressure depletion protocols under triaxial stress and uniaxial-strain boundary conditions. By interpolation of triaxial stress–strain data and adopting linear poroelastic theory, the uniaxial-strain compressibility required for the application of the Geertsma model is yielded. This does not require a priori information of elastic constants. Moreover, the interpolation technique allows for an independent check of the technical performance of the radial strain control system. The method holds for rocks that are close to linear poroelastic, subjected to stress conditions below the critical state line.

Bond stress distribution analysis between steel bar and steel fiber reinforced concrete using midpoint stress interpolation method

The bond stress distribution between steel bar and steel fiber reinforced concrete (SFRC) with different volume fractions were analyzed by the pullout test on the specimens with local bond, in which the high strength steel bar with inner strain gauge was embedded in SFRC. A new prediction model for bond stress was proposed, termed the midpoint stress interpolation method. The high accuracy of the proposed prediction model has been confirmed by comparison. Additionally, through analyzing the strains of steel bar, the distribution curves of bond stress under varying pullout loads were obtained. The results shown that the inhomogeneous distribution of bond stress between high strength steel bar and SFRC increases with the addition of steel fiber into high strength concrete. As the increase of steel fiber volume fraction, the slippage at the loaded end has a decreasing trend, and the bond stiffness near the loaded end increases greatly.

Computationally efficient viscoelastic flow simulation using a Lagrangian-Eulerian method and GPU-acceleration

A recently proposed Lagrangian-Eulerian method for viscoelastic flow simulation is extended to high performance calculations on the Graphics Processing Unit (GPU). The two most computationally intensive parts of the algorithm are implemented for GPU calculation, namely the integration of the viscoelastic constitutive equation at the Lagrangian nodes and the interpolation of the resulting stresses to the cell centers of the Eulerian grid.In the original CPU method, the constitutive equations are integrated with a second order backward differentiation formula, while with the proposed GPU method the implicit Euler method is used. To allow fair comparison, the latter is also implemented for the CPU. The methods are validated for two flows, a planar Poiseuille flow of an upper-convected Maxwell fluid and flow past a confined cylinder of a four-mode Phan Thien Tanner fluid, with identical results.The calculation times for the methods are compared for a range of grid resolutions and numbers of CPU threads, revealing a significant reduction of the calculation time for the proposed GPU method. As an example, the total simulation time is roughly halved compared to the original CPU method. The integration of the constitutive equation itself is reduced by a factor 50 to 250 and the unstructured stress interpolation by a factor 15 to 60, depending on the number of CPU threads used.