Node-based Smoothing Research Articles

A Limit Equilibrium Method Based on Geostress Calculated by Implicit Stabilized Node-Based Smoothed Finite Element Method for Stability Analysis of Soil Slopes

A Limit Equilibrium Method Based on Geostress Calculated by Implicit Stabilized Node-Based Smoothed Finite Element Method for Stability Analysis of Soil Slopes

A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

A Stable Node-Based Smoothed Finite Element Method with Transparent Boundary Conditions for the Elastic Wave Scattering by Obstacles

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of [Formula: see text] and [Formula: see text] semi-norm errors of SNS-FEM are faster.

A stable node-based smoothed finite element method with PML technique for the elastic wave obstacle scattering

In this paper, a stable node-based smoothed finite element method with PML (SNS-FEM-PML) is proposed to solve the scattering problem of a time-harmonic elastic plane wave by a rigid obstacle in two dimensions. In the algorithm, the stability term is constructed by the Taylor expansion of the gradient to cure the instability of the original NS-FEM. The linear variations of the gradient with respect to x and y are included in the stability term, which are calculated using four integral points in an equivalent circle of node-based smoothing domain. Meanwhile, the perfectly matched layer (PML) technique is used to truncate the unbounded domain. Furtherly, the smoothed Galerkin weak formulations of SNS-FEM-PML model are derived and the linear algebra system with the linear smoothed gradient is constructed for the Navier equation and Helmholtz equations with coupled boundaries. Besides, we also prove theoretically the softening effect and convergence of the SNS-FEM model. Several numerical examples verify the effectiveness and accuracy of SNS-FEM model. The results suggest that the convergence order of L2 and H1 semi-norm errors of the SNS-FEM model is consistent with the theory of FEM, and convergence rate of the relative error is higher than that of the FEM.

Contact Analysis Based on a Linear Strain Node-Based Smoothed Finite Element Method with Linear Complementarity Formulations

This paper presents the node-based smoothed finite element method with linear strain functions (NS-FEM-L) for solving contact problems using triangular elements. The smoothed strains are formulated by a complete order of polynomial functions and normalized with reference to the central points of smoothing domains. They are one order higher than those adopted in the finite element method (FEM) and the standard smoothed finite element method with the same triangular mesh. When using linear functions to describe strains in smoothing domains, the solutions are more accurate and stable. The contact interfaces are discretized by contact point pairs using a modified Coulomb frictional contact model. The contact problems are solved via converting into linear complementarity problems (LCPs) which can be tackled by using the Lemke method. Numerical implementations are conducted to simulate the contact behavior, including the bonding–debonding, contacting–departing and sticking–slipping. The effects of various parameters related to friction and adhesion are intensively investigated. The comparison of numerical results produced by different methods demonstrates the validity and efficiency of the NS-FEM-L for contact problems.

Diagonal Node-based Path Planning and Path Smoothing for First Responders and Rescue Robots

Diagonal Node-based Path Planning and Path Smoothing for First Responders and Rescue Robots

A stable node-based smoothed finite element method for stability analysis of two circular tunnels at different depths in cohesive-frictional soils

This paper investigates the stability of two circular tunnels at different depths in cohesive-frictional soils subjected to surcharge loading using a stable node-based smoothed finite element method (SNS-FEM). In the present method, the numerical integration domains are approximately circular regions of the node-based smoothing domain by the node-based smoothed finite element method (NS-FEM). Four additional integration points are employed for each node to form the stabilization items associated with the variance of the smoothed shape function gradient. The tunnels are modelled under plane strain conditions, and the soil is described as a uniform Mohr-Coulomb material, and it obeys an associated flow rule. The stability numbers of two circular tunnels are obtained directly by solving the optimization problems. In this study, the effects of soil properties (γD/c), the ratio of tunnel diameter to its depth (H/D), the horizontal and vertical spacing ratio (S/D, L/D) and soil internal friction angle (ϕ) on the stability numbers (σs/c) are investigated. Several numerical results of two circular tunnels have been carried out showing that the proposed approach can demonstrate the efficiency and reliability solutions. Based on upper bound limit analysis using SNS-FEM, the results are presented in the form of design tables and charts for engineers to use.

Stable node-based smoothed extended finite element method for fracture analysis of structures

Based on low-order elements, a stable node-based smoothed extended finite element method (SNS-XFEM) is proposed for fracture analysis of structures in this study. For the proposed method, the problem domain is discretized using low-order elements, which can be easily generated for structures with complex shapes. The node-based smoothing domains are then generated to perform the strain smoothing technique, which can effectively avoid singular integration. The discontinuity caused by crack is modeled using enrichment functions and a stabilization term based on strain gradient is also taken into account to further improve the accuracy. Finally, some numerical cases are studied to fully investigate the performance of present method. The obtained results show that the proposed SNS-XFEM can perform much better than standard XFEM and NS-XFEM.

A novel stabilized node-based smoothed radial point interpolation method (SNS-RPIM) for coupling analysis of magneto-electro-elastic structures in hygrothermal environment

In this article, a stabilized node-based smoothed radial point interpolation method (SNS-RPIM) was proposed to simulate the magneto-electro-elastic (MEE) structures in hygrothermal environment. The node-based smoothing domains were constructed based on triangular background elements, and the approximate strain, electric field intensity and magnetic flux density were derived using node-based strain smoothing operation. In the present formulation, the stabilization term related to the gradient variance of the field variables was used to obtain the system stiffness matrices. Then, by using the smoothed Galerkin weak form, discrete equations of the system were established to solve the generalized displacement of the structure. The numerical experiments validated the feasibility of SNS-RPIM for solving hygrothermo-magneto-electro-elastic (HMEE) coupling problems with different hygrothermal loadings and boundary conditions. Results proved that the introduction of the stabilization term made the SNS-RPIM had a ‘close-to-exact’ stiffness, thereby solving the ‘overly-soft’ in the traditional node-based smoothed radial point interpolation method (NS-RPIM). SNS-RPIM was in good agreement with the reference solution when using the same fewer nodes as finite element method (FEM), and the errors were much smaller than those of FEM. Moreover, SNS-RPIM performed better than NS-RPIM and FEM when processing distorted meshes. Therefore, the developed method showed great potential in simulating MEE structure in hygrothermal environment.

A four-node C$$^{0}$$ tetrahedral element based on the node-based smoothing technique for the modified couple stress theory

In this paper, a four-node $$C^{0}$$ tetrahedral element for the modified couple stress theory is proposed. Since the governing equations are the fourth-order differential equations, the first-order derivative of displacement or rotation should be approximated by a continuous function. In the proposed element, nodal rotations are defined using the node-based smoothing technique. Continuous rotation fields are defined with the shape functions and nodal rotations. Both the displacement field and the rotation field are expressed solely in terms of the displacement degrees of freedom. The element stiffness matrix is calculated using the newly defined rotation field. To prevent the increase of calculation cost due to increase of the bandwidth of the stiffness matrix, the preconditioned conjugate gradient method is introduced. The performance of the proposed element is evaluated through various numerical examples.

A node-based smoothed radial point interpolation method with linear strain fields for vibration analysis of solids

This paper presents a high-order node-based smoothed radial point interpolation method (NS-RPIM) with linear strain fields in smoothing domains. The linear smoothed strains are constructed by complete order of polynomial functions and normalized with reference to the central point of the smoothing region. The new NS-RPIM is one order higher than those of the existing methods which use piecewise constant strains. This high-order method still uses linear displacements within each triangular background cell, but linear strains are created over smoothing domains using the pick-out theory. Because the smoothed strain and the compatible strain within a local region are equal in an integral sense, the unknown parameters in the reconstructed strain functions can be determined uniquely by using three linearly independent weight functions. The numerical verification and computational efficiency are investigated and compared with the standard node-based smoothing models. It is found that the linear strain NS-RPIM shows improvement on both the convergence and computational efficiency in terms of error norm in displacement and is temporally stable. Since no spurious non-zero energy mode appears, this approach can be employed directly for the vibration analysis of solids.

Cell- and Node-Based Smoothing MITC3-Finite Elements for Static and Free Vibration Analysis of Laminated Composite and Functionally Graded Plates

Cell- and node-based smoothing MITC3-finite elements for static and free vibration analysis of laminated composite and functionally graded plates are proposed in this paper. It is based on a higher-order shear deformation theory which accounts for a higher-order variation of in-plane displacements and traction-free boundary conditions on top and bottom surfaces of transverse shear stresses. Novel CS-MITC3 and NS-MITC3 finite elements are proposed in which an MITC3 three-node triangle element with seven degrees-of-freedom per nodes that only requires the [Formula: see text]-type continuity is developed, and then cell- and node-based smoothing techniques are applied. Numerical results for the laminated composite and functionally graded plates are proposed to validate the accuracy of developed theory and to investigate the effects of material distribution and side-to-thickness ratio on the deflection, stresses and natural frequencies of the plates.

Highly accurate smoothed finite element methods based on simplified eight-noded hexahedron elements

Compared with the tetrahedron elements, hexahedron elements are preferred for their high accuracy. However, coordinate mapping required in the hexahedron elements of FEM formulation costs huge running time, leading to poor performance. Besides, the high quality of Jacobian matrix and mesh is required, which affects the accuracy of the strain results greatly. In order to solve these problems, we propose a novel simplified integration technique based on the smoothed finite element method (S-FEM) for the eight-noded hexahedron elements, where coordinate mapping is not demanded. The proposed new S-FEM-H8 models include simplified NS-FEM-H8 (using node-based smoothing domains) and simplified FS-FEM-H8 (using face-based smoothing domains). In the work, we divide a quadrilateral surface segment of a smoothing domain into two triangular sub-segments, so that the strain-displacement matrix can be calculated using a simple summation in the S-FEM theory instead of the integration in FEM. Then we conduct the Gauss integration scheme in each triangular surface sub-segment in order to avoid the coordinate mapping required in quadrilateral surface segments. The rest solving algorithm is the same as the standard S-FEM. Intensive numerical examples demonstrate that the simplified S-FEM-H8 possess the following features: (1) The strain energy of simplified NS-FEM-H8 is an upper bound of the exact solutions; (2) The simplified NS-FEM-H8 can overcome the volume locking problems for incompressible materials; (3) The method of dividing boundary surface into two triangular surfaces in smoothing domain keeps nearly the same accuracy as the standard S-FEM-H8.

A Hybrid Smoothed Finite Element Method for Predicting the Sound Field in the Enclosure with High Wave Numbers

Wave‐based methods for acoustic simulations within enclosures suffer the numerical dispersion and then usually have evident dispersion error for problems with high wave numbers. To improve the upper limit of calculating frequency for 3D problems, a hybrid smoothed finite element method (hybrid SFEM) is proposed in this paper. This method employs the smoothing technique to realize the reduction of the numerical dispersion. By constructing a type of mixed smoothing domain, the traditional node‐based and face‐based smoothing techniques are mixed in the hybrid SFEM to give a more accurate stiffness matrix, which is widely believed to be the ultimate cause for the numerical dispersion error. The numerical examples demonstrate that the hybrid SFEM has better accuracy than the standard FEM and traditional smoothed FEMs under the condition of the same basic elements. Moreover, the hybrid SFEM also has good performance on the computational efficiency. A convergence experiment shows that it costs less time than other comparison methods to achieve the same computational accuracy.

A stable node-based smoothed finite element method for metal forming analysis

In this paper, a stable node-based smoothed finite element method (SNS-FEM) is presented for analyzing metal forming problems using linear triangular or tetrahedral elements. In present method, the numerical integration domains are approximately circular or spherical regions of the node-based smoothing domains generated by the node-based smoothed finite element method (NS-FEM). Four or six supplementary integration points, which are symmetrically located at the crossover points of the region and the coordinate axis, are employed for each node to form the stabilization items associated with the variance of smoothed shape function gradient. Through this operation without the introducing of any uncertain parameter, the SNS-FEM not only significantly cures the temporal instability of NS-FEM but also performs better in effectiveness and efficiency than FEM, which is well validated by several numerical examples containing benchmark cases. Additionally, a simple but effective contact algorithm including contact searching and contact force computation is presented.

A node-based Smoothed Finite Element Method for Stability Analysis of Dual Square Tunnels in Cohesive-frictional Soils

This paper presents an upper bound limit analysis procedure using the node-based smoothed finite element method (NS-FEM) and second order cone programming (SOCP) to evaluate the stability of dual square tunnels in cohesive-frictional soils subjected to surcharge loading. The displacement field of the tunnel problems is approximated by using NS-FEM triangular elements (NS-FEM-T3). Next, commercial software Mosek is employed to deal with optimization problems, which are formulated as second order cone. Collapse loads and failure mechanisms of dual square tunnels were performed by solving the optimization problems with a series of size-to-depth ratios and soil properties. For dual square tunnels, the distance between centers of two parallel tunnels is the major parameter used to determine the stability. In this study, surcharge loading is applied to the ground surface and drained conditions are considered. Numerical results are verified with those available to demonstrate the accuracy of the proposed method.

High precision solution for thermo-elastic equations using stable node-based smoothed finite element method

The stable node-based smoothed finite element method (SNS-FEM) is presented to analyze thermo-elastic equations with various boundary conditions. The node-based smoothing domains are constructed to implement numerical integrations and then, the smoothed Galerkin weak form is utilized to obtain the discretized system equations. In the formulation, both the smoothed strains and the strain variance items over the node-based smoothing domains are used to perform the integration. Several numerical examples are studied in detail to investigate the performance of the SNS-FEM by comparison with the original NS-FEM and traditional FEM. Results show that the present SNS-FEM can provide very high precision solution for thermo-elastic equations. In addition, the SNS-FEM is more efficient than original NS-FEM and the standard FEM.

A stable nodal integration method for static and quasi-static electromagnetic field computation

A stable nodal integration method (SNIM) is presented to solve static and quasi-static electromagnetic problems in this paper. The analysis domain is firstly discretized into a set of triangular or tetrahedral elements, and linear interpolation is adopted within each element. A weakened weak formulation based on the nodes is further considered, framing the so-called node-based smoothing domains. Equivalent smoothing domains are then acquired as circular or spherical regions, where the gradient of shape function is expanded as the first order Taylor form. Subsequently, four or six temporary integration points on the region are picked to obtain items of the stiffness matrix and the external load vector. By simplifying the equations, the stiffness matrix can be received in quite concise form with one point integration and stabilization terms, which are calculated on original node-based smoothing domains. The implementation of SNIM on electromagnetic problems is thus realized. The proposed formulation is validated against both analytical solutions and traditional methods, and its effectiveness and potentialities can be well represented and clarified by numerical examples.

A generalized beta finite element method with coupled smoothing techniques for solid mechanics

This paper presents a generalized smoothing techniques based beta finite element method (βFEM) to improve the performance of standard FEM and the existing smoothed finite element methods (S-FEM) in solid mechanics. As we know, the edge-based (for 2D) or face-based (for 3D) strain smoothing techniques can bring much more accurate solutions than standard FEM, and offer lower bounds for force driven problems. The node-based smoothing technique with “overly-soft” feature, on the other hand has a unique property of producing upper bound solutions. This work proposes a novel generalized S-FEM with the smoothing domains generated based on both edges/faces and nodes. An adjustable parameter β is introduced to control the ratio of the area of edge/face-based and node-based smoothing domains. It is found that nearly exact solutions in strain energy can be obtained by tuning the parameter, making use of the important property that the exact solution is bonded by the solutions of NS-FEM and ES/FS-FEM. Standard patch tests are likewise satisfied. A number of numerical examples (static, dynamic, linear and nonlinear) have shown that the present βFEM method is found to be ultra-accurate, insensitive to mesh quality, temporal stable, capable of modeling complex geometry, immune from volumetric locking, etc.

Topology Optimization Using Node-Based Smoothed Finite Element Method

The node-based smoothed finite element method (NS-FEM) proposed recently has shown very good properties in solid mechanics, such as providing much better gradient solutions. In this paper, the topology optimization design of the continuum structures under static load is formulated on the basis of NS-FEM. As the node-based smoothing domain is the sub-unit of assembling stiffness matrix in the NS-FEM, the relative density of node-based smoothing domains serves as design variables. In this formulation, the compliance minimization is considered as an objective function, and the topology optimization model is developed using the solid isotropic material with penalization (SIMP) interpolation scheme. The topology optimization problem is then solved by the optimality criteria (OC) method. Finally, the feasibility and efficiency of the proposed method are illustrated with both 2D and 3D examples that are widely used in the topology optimization design.