Cell-based Smoothed Finite Element Method Research Articles

A reduced smoothed integration scheme of the cell‐based smoothed finite element method for solving fluid–structure interaction on severely distorted meshes

AbstractThis article describes an inexpensive partitioned coupling strategy for computational fluid–structure interaction (FSI) admitting negative‐Jacobian elements. The emphasis is very much on a reduced smoothed integration (RSI) scheme of the cell‐based smoothed finite element method (CSFEM) using four‐node quadrilateral (Q4) elements for a cost‐effective solution to the Navier–Stokes (NS) equations. In the discrete fluid field, each Q4 element is considered as one single smoothing cell so as to diminish the smoothed integration loops substantially. However, the RSI scheme does not respect the stability condition of smoothed Galerkin weak‐form integral in the CSFEM. To tackle this issue, a simple hourglass control is introduced to the under‐integrated formulation of the NS solver. Importantly, the stabilized RSI scheme has an inbuilt advantage of its enormous tolerance towards negative‐Jacobian elements. The developed technique is easy‐to‐implement and has been tested in various FSI examples adopting both fine and distorted meshes.

A cell-based smoothed finite element model for the analysis of turbulent flow using realizable k-ε model and mixed meshes

Smoothed finite element method (S-FEM) has been widely employed in computational mechanics, particularly in computational solid mechanics (CSM) and, to a lesser extent, in computational fluid dynamics (CFD). The present work focuses on the development of S-FEM for solving three-dimensional incompressible turbulent flow problems using the realizable k-ε model. The Streamline Upwind Petrov-Galerkin approach combined with Stabilized Pressure Gradient Projection (SUPG/SPGP) was used to restrain/control the spatial oscillation and instability in conjunction with mixed meshes employed to discretize complex geometries. The presented turbulent cell-based S-FEM (CS-FEM) was validated under several common flow types, including mesh turbulence, turbulent channel flow, turbulent flow past a circular cylinder, flow separation around an obstacle, flow in the dimpled channel, and flow in the upper human airway. In particular, the presented work exhibited the latent capability of the CS-FEM in solving turbulent flows near boundary regions and strong convection regions. Meanwhile, the results of the CS-FEM were compared to those of the lowest equal-order finite element method (i.e. P1–P1 or Q1–Q1 elements) in simulating turbulent flows, and it was found that the CS-FEM had better agreement with other published works.

Quasi-brittle and Brittle Fracture Simulation Using Phase-field Method based on Cell-based Smoothed Finite Element Method

Quasi-brittle and Brittle Fracture Simulation Using Phase-field Method based on Cell-based Smoothed Finite Element Method

A cell-based smoothed finite element method for multi-body contact analysis within the bi-potential formulation

This paper presents a cell-based smoothed finite element method (CS-FEM) for solving contact problems within the bi-potential formulation. The frictional contact and large deformation are modeled based on the CS-FEM. This method performs based on the generalized smoothed Galerkin weak-form which requires only value of shape functions. Because no isoparametric mapping is needed, the CS-FEM has an advantage when dealing with problems with large deformation which may cause mesh distortion. The contact forces and the relative velocity establish an implicit relationship within the bi-potential formulation in which no user-defined parameter is introduced. Numerical examples show the contact between elastic and hyperelastic bodies calculated by CS-FEM models with different smoothing domains. Compared with FEM, the CS-FEM produces robust and more accurate solutions.

N-Side Cell-Based Smoothed Finite Element Method for Incompressible Flow with Heat Transfer Problems

In this paper, an n-sided cell-based smoothed finite element method (nCS-FEM) is presented to solve the convective heat transfer in incompressible viscous fluid flow. The Characteristic-based-split (CBS) scheme is used to stabilize the spatial and pressure oscillations of CS-FEM that occur during the process of solving incompressible Navier-Stokes equations. Meanwhile, the characteristic-Galerkin method in CBS is adopted as the convective stabilization in the heat transfer equation. In order to verify the proposed method, typical numerical examples are used to test the computational accuracy of CS-FEM. Meanwhile, the investigation of the robustness of highly distorted element is conducted in a natural convection test case. The calculation results of numerical examples show that CS-FEM is qualified to solve the steady and unsteady convective heat transfer problems for incompressible laminar flow with high reliability. The flow across tube banks with complex geometry in the heat sinks is also simulated to prove the capabilities of the proposed method. Well agreed comparison with the results of CFD software, Fluent, proves that the present method can effectively simulate realistic and complex flow with heat transfer problems.

A cell-based smoothed finite element model for non-Newtonian blood flow

Smoothed Finite Element Method (S-FEM) has drawn increasing attention in the field of computational fluid dynamics (CFD) and the present work seeks to make further contribution to this growing field of S-FEM by simulating for the non-Newtonian blood flow. This investigation took the form of Streamline Upwind Petrov-Galerkin in conjunction with Stabilized Pressure Gradient Projection (SUPG/SPGP) to alleviate the spatial oscillation and instability problems. The validation of the cell-based S-FEM (CS-FEM) combined with SUPG/SPGP was carried out by the blood flow over a backward-facing step. The performances of the presented method were explored by the blood flow in the carotid bifurcation and blood flow in the intracranial segment of internal carotid artery. Impressively, the results exhibit good features of the CS-FEM on solving severely distorted mesh vis-a-vis the standard finite element method (FEM). The presented method could realize the accurate prediction in the mentioned complex blood flows as the same as Finite Volume Method software STAR-CCM+.

A stable CS-FEM for the static and seismic stability of a single square tunnel in the soil where the shear strength increases linearly with depth

A numerical procedure using a stable cell-based smoothed finite element method (CS-FEM) is presented for estimation of stability of a square tunnel in the soil where the shear strength increases linearly with depth. The kinematically admissible displacement fields are approximated by uniform quadrilateral elements in conjunction with the strain smoothing technique, eliminating volumetric locking issues and the singularity associated with the Mohr–Coulomb model. First, a rich set of simulations was performed to compute the static stability of a square tunnel with different geometries and soil conditions. The presented results are in excellent agreement with the upper and lower bound solutions using the standard finite element method (FEM). The stability charts and tables are given for practical use in the tunnel design, along with a newly proposed formulation for predicting the undrained stability of a single square tunnel. Second, the seismic stability number was computed using the present numerical approach. Numerical results reveal that the seismic stability number reduces with an increasing value of the horizontal seismic acceleration (αh), for both cases of the weightless soil and the soil with unit weight. Third, the link between the static and seismic stability numbers is described using corrective factors that represent reductions in the tunnel stability due to seismic loadings. It is shown from the numerical results that the corrective factor becomes larger as the unit weight of soil mass increases; however, the degree of the reduction in seismic stability number tends to reduce for the case of the homogeneous soil. Furthermore, this advanced numerical procedure is straightforward to extend to three-dimensional (3D) limit analysis and is readily applicable for the calculation of the stability of tunnels in highly anisotropic and heterogeneous soils which are often encountered in practice.

A coupled cell-based smoothed finite element method and discrete phase model for incompressible laminar flow with dilute solid particles

In this paper, the cell-based smoothed finite element method (CS-FEM) empowered by the discrete phase model (DPM) is developed to solve dilute solid particles movements induced by incompressible laminar flow. In the present method, the fluid phase is solved by CS-FEM in the Eulerian framework, while particles are treated as discrete phases traced using Newton's second law in the Lagrangian framework. Meanwhile, the fluidic drag force on particles is considered to realize the one-way coupling of fluid to particles. For the fluid phase, the semi-implicit characteristic-based split (CBS) method is employed to suppress the spatial and pressure oscillations arising from the numerical solution of the Navier-Stokes equations discretized by the CS-FEM. To accurately capture the fluid velocity at an arbitrary particle position inside quadrilateral elements, the mean value coordinates interpolation is introduced. Furthermore, the motion equations for particles are solved by the fourth-order Runge-Kutta method to ensure high accuracy on particle trajectories. Several numerical examples in this paper demonstrate that the proposed method can effectively predict the effect of fluid flow on particle trajectories and position distributions in the analysis of practical and complex flow problems.

A phase-field cohesive zone model integrated with cell-based smoothed finite element method for quasi-brittle fracture simulations of concrete at mesoscale

In this work, a phase field cohesive zone model (PFCZM) combined with the cell-based smoothed finite element method (CSFEM) is presented to investigate the quasi-brittle fracture behaviour of concrete at mesoscale. An automatic quadtree decomposition technique is employed to discretise the CT image-based meso-structures including aggregates, mortar and pores, with local mesh adaptively refined at material boundaries. The resultant quadtree elements with hanging nodes are naturally treated as CSFEM polygons. In the proposed CSFEM-PFCZM, the stiffness matrices and residual vectors for displacement and damage fields are formulated by the Wachspress shape function and smoothed strain matrices. The crack driving forces are evaluated as history variables at the integration points of CSFEM subcells. Four nonlinear fracture benchmarks are presented to validate the proposed method regarding the variability of mesoscale fracture processes, crack topology and load–displacement curves. It is found that the CSFEM-PFCZM combined with the quadtree decomposition significantly improves the computational efficiency over the traditional FEM counterparts, and shows great potential for multiscale fracture evaluation of quasi-brittle composite materials. This is due to the fast mesh generation and flexible simulation of complicated nonlinear fracture using the CSFEM-PFCZM without the need for remeshing or mesh enrichments.

A Cell-Based Smoothed Finite Element Method for Solving Incompressible Reynolds-Averaged Navier–Stokes Equations Using Spalart–Allmaras Turbulence Model

In this paper, a leap forward in the development of a cell-based smoothed finite element method (CS-FEM) is presented to solve incompressible turbulent flow treated by Reynolds–averaged Navier–Stokes (RANS) equations and the Spalart–Allmaras (SA) model. The streamline-upwind/Petrov–Galerkin stabilization combined with the stabilized pressure gradient projection (SUPG/SPGP) is used to address oscillation and instability for CS-FEM in calculations of incompressible flow. The capability, accuracy, and validity of presented turbulent CS-FEM were tested by four numerical examples, including turbulent channel flow, a backward-facing step, flow past a circular cylinder, and turbulent flow past multistage Tesla valves. Based on the favorable comparison of numerical results obtained by the presented CS-FEM with experimental results and literature numerical results, CS-FEM is demonstrated to be highly applicable with appealing performances on simulations of incompressible turbulent flows. This new extension of CS-FEM empowers the S-FEM family to solve complex turbulent flows.

A stabilized cell‐based smoothed finite element method against severe mesh distortion in non‐Newtonian fluid–structure interaction

AbstractFailure is an inevitability for the finite element method (FEM) to be performed on seriously degenerated four‐node quadrilateral (Q4) elements whose Jacobians become negative. This article proposes a stabilized selective integration formulation of the cell‐based smoothed FEM (CSFEM) for modeling non‐Newtonian fluid–structure interaction (NNFSI) where that acute mesh distortion arises. The Carreau–Yasuda and Oldroyd‐B fluids, respectively, interact with a geometrically nonlinear solid in NNFSI. The CSFEM is applied to discretize both the fluid and solid media in space. As the fluid mesh accounts for a substantial part of the discretized NNFSI system, seriously distorted Q4 elements are specified exclusively for the fluid field. In this case, each convex Q4 element is subdivided into four smoothing cells (SCs) whereas its concave counterpart is regarded as one single SC. In the meantime, the solid elements are treated as usual. To stabilize the one‐SC integral in smoothed Galerkin weak form, an hourglass control is subsequently introduced into the fluid momentum equations as well as the viscoelastic constitutive equation. The transient non‐Newtonian fluid equations are advanced with dual time steps of the characteristic‐based split scheme in time to further enhance the numerical stability and accuracy. After discussing other aspects of computational NNFSI, two benchmark problems are presented to demonstrate the effectiveness and performance of the developed methodology under the harsh environment.

Cell-Based Smoothed Finite-Element Framework for Strongly Coupled Non-Newtonian Fluid–Structure Interaction

AbstractThis work presents a cell-based smoothed finite-element method (CS-FEM) for simulating strongly coupled non-Newtonian fluid–structure interactions. The governing equations of a Carreau-Yasu...

A cell-based smoothed finite element method (CS-FEM) for three-dimensional incompressible laminar flows using mixed wedge-hexahedral element

Smoothed finite element method (S-FEM) has attracted lots of attentions in the fields of computational mechanics, especially in solid mechanics and heat transfer problems. In computational fluid dynamics, works on S-FEM were limited to two-dimensional problems. This work aims to extend the S-FEM to three-dimensional (3D) incompressible laminar flows. Wedge element grids and grids with mixed wedge and hexahedral elements are formulated for 3D incompressible laminar flows based on the cell-based S-FEM (CS-FEM). To reduce numerical oscillations, we implemented the streamline-upwind/Petrov-Galerkin method (SUPG) together with the stabilized pressure gradient projection (SPGP). Several examples are presented, including the Beltrami flow, lid-driven cavity flow, backward facing step flow and microchannel flow, to validate and examine the presented method. The results indicate that wedge elements and mixed wedge-hexahedral elements based on the CS-FEM have higher computational efficiency than that of hexahedral elements based on the CS-FEM for the same level of computational accuracy. It is also found that the present CS-FEM performed better than the standard FEM in dealing with pressure stability. The flow characteristics are well captured by the CS-FEM using the mixed wedge-hexahedral elements, and the numerical results are acceptable compared to those of STAR-CCM+.

A phase-field modelling for 3D fracture in elasto-plastic solids based on the cell-based smoothed finite element method

Three-dimensional (3D) phase-field models based on the cell-based smooth finite element method (CS-FEM) are established in this paper. The present model consists of three key ingredients: 1) a phase-field model of fracture in elasto-plastic solids, 2) a staggered scheme for nonlinear interactions between the phase-field equations and those for the elasto-plastic solid mechanics, and 3) its underline solution platform of the cell-based smoothed finite element method, CS-FEM. The phase-field model allows effective capturing of the evolution process of complex fracture morphology in elasto-plastic solids. The use of the staggered algorithm gives a practical means to couple the phase field and displacement field in fracturing elasto-plastic solids. The CS-FEM offers important softer model behavior in solving the governing equations. For wide applicability of our Phase-field CS-FEM approach, it is implemented in the commercial software ABAQUS via User Defined Element (UEL) subroutine. Numerical examples of both 2D and 3D benchmark problems with complex crack topologies are presented to verify the accuracy of the solutions. Good agreements are obtained with the experimental observations and existing numerical results in the literature. The phase-field CS-FEM approach offers an effective alternative means to model 3D fracturing in elasto-plastic solids.

A multi-physics coupling cell-based smoothed finite element micromechanical model for the transient response of magneto-electro-elastic structures with the asymptotic homogenization method

In this paper, the Cell-based smoothed finite element method (CS-FEM) and the asymptotic homogenization method (AHM) are used to accurately simulate the responses of 1–3 type magneto-electro-elastic (MEE) structure under dynamic load. The dynamic characteristics of micromechanics are discussed. Firstly, the properties of MEE materials at different composition ratios are calculated by AHM. In the next step, the CS-FEM equations for the multi-physics problems are deduced based on the MEE constitutive equations and the effective parameters. The transient reactions of the problem are solved by introducing the modified Wilson-θ approach. To illustrate the precision, reliability and convergence of CS-FEM, four numerical examples of various thin-walled structures are designed, results of CS-FEM are compared with those of the traditional finite element method (FEM) with denser elements. It shows that CS-FEM has higher accuracy, faster convergence speed and higher computational efficiency. This paper has some application value for the design and development of thin-walled intelligent sensors and energy harvesters, etc.

A high order cell-based smoothed finite element method using triangular and quadrilateral elements

This paper presents a novel high-order cell-based smoothed finite element method (CS-FEM) using 6 nodes triangular (T6) and 8 nodes quadrilateral (Q8) elements. In addition, high-order quadrilateral transition elements (Q5 and Q6) are also developed based on this method. In this method, the high order strain field in each smoothing domain is constructed using pick-out theory. And the strain field is represented by a complete polynomial. Both triangular and quadrilateral high order CS-FEM are formulated without the need to dividing the elements into smaller smoothing cells. Since there is no mapping and derivation in computation, so high-order CS-FEM has better adaptability to mesh distortion. The performance of high-order CS-FEM is examined in great detail, using a number of numerical examples. And it is found that CS-FEM results agrees with finite element method (FEM) counterpart. In addition, we found also that the edge-nodes of the high-order elements in a CS-FEM model can be quite freely placed, which is not possible in a standard high-order FEM model. All the high-order CS-FEM models are implemented through User-defined Element Library (UEL) in ABAQUS, which greatly improves work efficiency.

Stabilization of a smoothed finite element semi-implicit coupling scheme for viscoelastic fluid–structure interaction

We propose in this work a stabilization approach for semi-implicit coupling of viscoelastic fluid–structure interaction (VFSI) using the cell-based smoothed finite element method (CS-FEM). The viscoelastic fluid and nonlinear solid equations are spatially discretized by the CS-FEM and then are semi-implicitly coupled via a partitioned solution strategy. The current semi-implicit coupling framework depends on a second-order characteristic-based split (CBS(B)) scheme that solves the Navier–Stokes equations together with the Oldroyd-B constitutive model in the fractional-step manner. To enhance the stability of the semi-implicit coupling algorithm, the discrete elastic-viscous split stress-gradient (DEVSS-G) procedure is introduced into the explicit stage while the stabilized pressure gradient projection (SPGP) is earmarked for the implicit stage. Moreover, the iterated end-of-step velocity begins with the intermediate velocity during the subiterations. The DEVSS-G/CBS(B)-SPGP technique is readily applied to the CBS-based partitioned semi-implicit coupling algorithm for VFSI. Visible improvements in stabilization and efficiency are revealed in a benchmark test. • A semi-implicit coupling scheme is proposed for viscoelastic fluid–structure interaction. • DEVSS-G/CBS(B)-SPGP stabilization is presented for the semi-implicit scheme. • VFSI system is entirely formulated by cell-based smoothed finite element method. • The developed technique boosts the stability and efficiency considerably.

Contact Analysis Within the Bi-Potential Framework Using Cell-Based Smoothed Finite Element Method

This paper presents a cell-based smoothed finite element method (CS-FEM) for solving two-dimensional contact problems with the bi-potential formulation. The contact force and the relative displacements on the contact surface are coupled with each other. The Uzawa algorithm, which is a local iterative technique, is used to solve the contact force. The classic Coulomb friction rule and a unilateral contact relationship are considered. There is no need to select any user-defined parameter in the whole process. Three contact states are investigated accurately, which can be stated as sticking, separating and sliding, respectively. The CS-FEM is performed with six different kinds of smoothing domains which are constructed by dividing the background element into different regions. Only boundary integrations instead of domain integral are required in the calculation, and no coordinate mapping is needed. Three numerical examples are presented to verify the effectiveness of the method. The effect of the friction coefficient for the contact is also investigated. All the obtained numerical solutions agree well with the reference values. The results produced by the CS-FEM are more accurate than those of the traditional FEM. Moreover, the CS-FEM can provide both an upper bound and a lower bound of the strain energy solutions while using different smoothing domains.

Dynamic Response of the Piezoelectric Materials Based on Cell-Based Smoothed Finite Element Method in Hygrothermal Environment

In this paper, the dynamic response of piezoelectric structures under hygrothermal environment is studied by using cell-based smoothed finite element method (CS-FEM). Ignoring the influence of temperature and moisture on the response of elastic matrix, we derive the basic equations of the piezoelectric materials under hygrothermal environment. Then the CS-FEM equations of piezoelectric problem are deduced based on the constitutive equation of the material. The improved Newmark scheme is used to solve the transient response of this problem. Several numerical examples are given to illustrate the advantage of CS-FEM in dealing with piezoelectric materials in hygrothermal environment. Results of this study provides some ideas for the design and manufacture of piezoelectric smart structures in hygrothermal environment.

A Cell-Based Smoothed Finite Element Method for Arbitrary Polygonal Element to Solve Incompressible Laminar Flow

In this paper, a cell-based smoothed finite element method using the arbitrary [Formula: see text]-sided polygonal element (CS-FEM-Poly) is developed to solve fluid mechanics problems. A stabilization method, characteristic-based split coupled with stabilized pressure gradient projection (CBS/SPGP), is employed to deal with numerical oscillations for CS-FEM-Poly. We validate CS-FEM-Poly and test its numerical behaviors using triangular, quadrilateral and polygonal elements on four typical numerical examples. Numerical results show that the CS-FEM-Poly based on CBS/SPGP produces well-agreed solutions with the exact solutions of benchmarks, and gives desirable convergence rate as compared with FEM. In the backward-facing step flow example, the numerical robustness for concave elements is manifested for CS-FEM-Poly. The proposed CS-FEM-Poly exhibits the remarkable potential for polygonal mesh or glued polygonal elements in the hybrid mesh to solve incompressible flows.