Background Mesh Research Articles

A stable discontinuity-enriched finite element method for 3-D problems containing weak and strong discontinuities

A new enriched finite element technique, named the Discontinuity-Enriched Finite Element Method (DE-FEM), was introduced recently for solving problems with both weak and strong discontinuities in 2-D. In this mesh-independent procedure, enriched degrees of freedom are added to new nodes collocated at the intersections between discontinuities and the sides of finite elements of the background mesh. In this work we extend DE-FEM to 3-D and describe in detail the implementation of a geometric engine capable of handling interactions between discontinuities and the background mesh. Several numerical examples in linear elastic fracture mechanics demonstrate the capability and performance of DE-FEM in handling discontinuities in a fully mesh-independent manner. We compare convergence properties and the ability to extract stress intensity factors with standard FEM. Most importantly, we show DE-FEM provides a stable formulation with regard to the condition number of the resulting system stiffness matrix.

Immersogeometric analysis of moving objects in incompressible flows

We deploy the immersogeometric approach for tracking moving objects. The method immerses objects into non-boundary-fitted meshes and weakly enforces Dirichlet boundary conditions on the object boundaries. The object motion is driven by the integrated surface force and external body forces. A residual-based variational multiscale method is employed to stabilize the finite element formulation for incompressible flows. Adaptively refined quadrature rules are used to better capture the geometry of the immersed boundaries by accurately integrating the intersected background elements. Treatment for the freshly-cleared nodes (i.e. background mesh nodes that are inside the object at one time step, but are in the fluid domain at the next time step) is considered. We assess the accuracy of the method by analyzing object motion in different flow structures including objects freely dropping in viscous fluids and particle focusing in unobstructed and obstructed micro-channels. We show that key quantities of interest are in very good agreements with analytical, numerical and experimental solutions. We also show a much better computational efficiency of this framework than current commercial codes using adaptive boundary-fitted approaches. We anticipate deploying this framework for applications of particle inertial migration in microfluidic channels.

An improved time domain meshfree method for analysis of quasi-static and dynamic inhomogeneous viscoelastic problems

An improved meshfree radial point interpolation method (RPIM) is formulated for quasi-static and dynamic analyses of 2D inhomogeneous viscoelastic solids. The presented improved RPIM is formulated in time domain, in which, there is no need for the viscoelastic material to obey the correspondence principle and therefore, all kinds of inhomogeneous viscoelastic materials can be analyzed by the method. The domain integrals, appeared in the proposed RPIM, are computed by the Cartesian transformation method (CTM), which is a truly meshfree integration technique. In the implementation of the CTM no background mesh is required and the domain integrals are computed accurately and efficiently. The accuracy of the method is studied by presenting several numerical examples. The results are compared with available analytic or numerical solutions and very good agreement is observed in all cases.

Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

AbstractIntegro‐differential equations with non‐integer order derivatives are an all‐purpose subdivision of fractional calculus. In the current paper, we present a numerical method for solving fractional Volterra‐Fredholm integro‐differential equations of the second kind. To establish the scheme, we first convert these types of integro‐differential equations to mixed integral equations by fractional integrating from both sides of them. Then, the discrete collocation method by combining the locally supported radial basis functions are used to approximate the mentioned integral equations. Since the local method proposed in this paper estimates an unknown function via a small set of data instead of all points in the solution domain, it uses much less computer memory in comparison with the global cases. We apply the nonuniform Gauss‐Legendre integration rule to compute the singular‐fractional integrals appearing in the scheme. As the scheme does not need any background meshes, it can be recognized as a meshless method. Numerical results are included to show the validity and efficiency of the new technique and confirm that the algorithm of the presented approach is attractive and easy to implement on computers.

Fully coupled elastoplastic hydro‐mechanical analysis of unsaturated porous media using a meshfree method

SummaryThis paper presents the first application of an advanced meshfree method, ie, the edge‐based smoothed point interpolation method (ESPIM), in simulation of the coupled hydro‐mechanical behaviour of unsaturated porous media. In the proposed technique, the problem domain is spatially discretised using a triangular background mesh, and the polynomial point interpolation method combined with a simple node selection scheme is adopted for creating nodal shape functions. Smoothing domains are formed on top of the background mesh, and a constant smoothed strain, created by applying the smoothing operation over the smoothing domains, is assigned to each smoothing domain. The deformation and flow models are developed based on the equilibrium equation of the mixture, and linear momentum and mass balance equations of the fluid phases, respectively. The effective stress approach is followed to account for the coupling between the flow and deformation models. Further coupling among the phases is captured through a hysteretic soil water retention model that evolves with changes in void ratio. An advanced elastoplastic constitutive model within the context of the bounding surface plasticity theory is employed for predicting the nonlinear behaviour of soil skeleton. Time discretisation is performed by adopting a three‐point discretisation method with growing time steps to avoid temporal instabilities. A modified Newton‐Raphson framework is designed for dealing with nonlinearities of the discretised system of equations. The performance of the numerical model is examined through a number of numerical examples. The state‐of‐the‐art computational scheme developed is useful for simulation of geotechnical engineering problems involving unsaturated soils.

A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique

The current investigation studies a numerical method to solve Volterra integral equations of the second kind arising in the single term fractional differential equations with initial conditions. The proposed method estimates the solution of the mentioned Volterra integral equations using the discrete Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least squares polynomial fitting. We compute fractional integrals appeared in the method by a suitable integration rule based on the non-uniform composite Gauss-Legendre quadrature formula. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve fractional differential equations and its algorithm can be easily implemented on computers. The error bound and the convergence rate of the presented method are obtained. Finally, numerical examples are included to show the validity and efficiency of the new technique.

A cell‐based smoothed radial point interpolation method with virtual nodes for three‐dimensional mid‐frequency acoustic problems

SummaryIt is well known that the finite element method (FEM) encounters dispersion errors in coping with mid‐frequency acoustic problems due to its “overly stiff” nature. By introducing the generalized gradient smoothing technique and the idea of condensed shape functions with virtual nodes, a cell‐based smoothed radial point interpolation method is proposed to solve the Helmholtz equation for the purpose of reducing dispersion errors. With the properly selected virtual nodes, the proposed method can provide a close‐to‐exact stiffness of continuum, leading to a conspicuous decrease in dispersion errors and a significant improvement in accuracy. Numerical examples are examined using the present method by comparing with both the traditional FEM using four‐node tetrahedral elements (FEM‐T4) and the FEM model using eight‐node hexahedral elements with modified integration rules (MIR‐H8). The present cell‐based smoothed radial point interpolation method has been demonstrated to possess a number of superiorities, including the automatically generated tetrahedral background mesh, high computational efficiency, and insensitivity to mesh distortion, which make the method a good potential for practical analysis of acoustic problems.

Numerical Simulation of Seepage Flow through Dam Foundation Using Smooth Particle Hydrodynamics Method

In this paper, a mesh-free approach called smooth particle hydrodynamics (SPH) is proposed to analyze the seepage problem in porous media. In this method, computational domain is discredited by some nodes, and there is no need for background mesh; therefore, it is a truly meshless method. The method was applied to analyze seepage flow through a concrete dam foundation. Using the SPH method, the computational boundary being coincident with the physical boundary, was numerically acquired by solving seepage flow govern in equations. The numerical results of the presented method were compared with ones calculated by the Geostudio-SEEPW (finite element based software). The water head values were calculated through the dam foundation, and there was a good agreement between results. Moreover, results showed that the SPH method is efficient and capable of analyzing seepage flow particularly in complex geometry problems.

Manufacturing oriented topology optimization of 3D structures for carbon emission reduction in casting process

Low-carbon casting for box-beams is achieved by using an explicit topology optimization method that provides an optimized stiffener layout with fewer carbon emissions. First, a life-cycle assessment of carbon generated during casting stage is presented in detail. Considering the carbon assessment is formulated in an explicit form, an explicit topology optimization method is applied, where stiffeners are explicitly described with geometric parameters. Additionally, the stiffeners are able to move freely without background mesh constraints. This trait is achieved by an interpolation algorithm that spreads the stiffness of the stiffeners to adjacent background nodes. To illustrate this method, a press slide is investigated, and the optimized design shows a 17.6% reduction in carbon emissions with no reduction of stiffness.

Modified numerical approaches for a class of Volterra integral equations with proportional delays

This paper addresses modified-meshless numerical schemes for dynamical systems with proportional delays. The proposed mesh reduction techniques are based on a redial-point interpolation and moving least-squares methods. An optimal influence domain radius is constructed utilizing nodal connectivity and node-depending integration background mesh. Optimal shape parameters are obtained by the use of properties of the delta Kronecker and the compactly supported weight function. Numerical results are provided to justify the accuracy and efficiency of the proposed schemes.

Enhancement of pressure calculation in projection-based particle methods by incorporation of background mesh scheme

Particle methods, or Lagrangian mesh-free methods, are well known as potentially robust computational methods for a wide range of applications in science and engineering, including ocean engineering. Despite their potential robustness, a challenge in the application of particle methods corresponds to the presence of unphysical pressure noise. This study presents a simple and straightforward approach for enhancement of pressure calculation in projection-based particle methods, in the context of MPS (Moving Particle Semi-implicit) method. One of the causes for the presence of unphysical pressure noise in projection-based particle methods corresponds to the use of Eulerian kernels (with a fixed radius of influence or smoothing length) with a purely Lagrangian description of motions in the presence of particle perturbations, leading to spatial discontinuities in the source term of Poisson Pressure Equation (PPE). To enhance continuity, accuracy and smoothness of calculated source term of PPE, a so-called Background Mesh (BM) scheme is proposed to provide spatial connectivity and continuity in between calculated source terms at moving particle positions. The scheme is validated through several benchmark tests including a simple uniform flow, a dam-breaking flow and violent sloshing flows. Incorporation of BM scheme is shown to be effective in enhancing the pressure calculation in projection-based particle methods with only a slight increase in computational cost.

An isotropic unstructured mesh generation method based on a fluid relaxation analogy

In this paper, we propose an unstructured mesh generation method based on Lagrangian-particle fluid relaxation, imposing a global optimization strategy. With the presumption that the geometry can be described as a zero level set, an adaptive isotropic mesh is generated by three steps. First, three characteristic fields based on three modeling equations are computed to define the target mesh-vertex distribution, i.e. target feature-size function and density function. The modeling solutions are computed on a multi-resolution Cartesian background mesh. Second, with a target particle density and a local smoothing-length interpolated from the target field on the background mesh, a set of physically-motivated model equations is developed and solved by an adaptive-smoothing-length Smoothed Particle Hydrodynamics (SPH) method. The relaxed particle distribution conforms well with the target functions while maintaining isotropy and smoothness inherently. Third, a parallel fast Delaunay triangulation method is developed based on the observation that a set of neighboring particles generates a locally valid Voronoi diagram at the interior of the domain. The incompleteness of near domain boundaries is handled by enforcing a symmetry boundary condition. A set of two-dimensional test cases shows the feasibility of the method. Numerical results demonstrate that the proposed method produces high-quality globally optimized adaptive isotropic meshes even for high geometric complexity.

Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions

We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche’s method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

Dynamic load and stress analysis of a large horizontal axis wind turbine using full scale fluid-structure interaction simulation

A dynamic load and stress analysis of a wind turbine is carried out using transient fluid-structure interaction simulations. On the structural side, the three 50 m long commercial glass-fiber epoxy blades are modelled using shell elements, accurately including the properties of the composite materials. On the fluid side, a hexahedral mesh is obtained for every blade and for the hub of the machine. These meshes are then overlaid to a structured background mesh through an overset technique. The displacements prescribed by the structural solver are imposed on top of the rigid rotation of the turbine. The atmospheric boundary layer (ABL) is included using the k-epsilon turbulence model. The computational fluid dynamics (CFD) and computational solid mechanics (CSM) solvers are strongly coupled using an in-house code. The transient evolution of loads, stresses and displacements on each blade is monitored throughout the simulated time. The ABL induces oscillating axial displacements in the outboard region of the blade. Furthermore, the influence of gravity on the structure is accounted for and investigated, showing that it largely affects the tangential displacement of the blade. The oscillating deformations lead to sensible differences in the torque provided by each blade during its rotation.

An implicit boundary approach for viscous compressible high Reynolds flows using a hybrid remeshed particle hydrodynamics method

We present an implicit boundary particle method with background mesh adaptation. We use a Brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high Reynolds numbers. A penalty term is added to the Navier-Stokes equations to impose the boundary conditions. The boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid, achieving an implicit approach for flow around complex boundaries/geometries simulation. The main idea of the Brinkman penalisation method is to model the solid obstacle as a porous medium. The governing equations for the compressible fluid and penalised Navier-Stokes for the porous medium are solved simultaneously, without need for interface conditions. The accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel (Poiseuille flow), to more complex problems such as the lid-driven cavity with high Reynolds numbers, the reflection and propagation of a compressible shock wave, and finally swirling flow in a model two-stroke diesel engine. The remeshed particle-mesh method coupled with Brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions.

A cell-based smoothed radial point interpolation method using condensed shape functions for free and forced vibration analysis of solids

In this work, a novel cell-based smoothed radial point interpolation method (CS-RPIM) has been developed for free and forced vibration analysis of solids by using condensed radial point interpolation method (RPIM) shape functions with virtual points. In the scheme of the present method, the generalized gradient smoothing technique (GST) is used for constructing smoothed strains and the ingenious six virtual nodes are selected together with real nodes for creating condensed RPIM shape functions, which helps the proposed method avoid not only the overly-stiff problem existing in traditional finite element method (FEM) but also the overly-soft issue of original CS-RPIM models. Thanks to the well-constructed system stiffness, the proposed method has shown very good performance in terms of accuracy, efficiency and stability for the examples studied. As three-nodes triangular elements, that can be automatically generated for complex problem domains, are used as background mesh, the proposed method is supposed to possess good potential for solving practical engineering problems.

An Eulerian finite element method for PDEs in time-dependent domains

The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.

Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture

Efficient implementation of the element-free Galerkin (EFG) method for a phase-field model of linear elastic fracture mechanics is presented, in which the convenience of the meshfree method to construct high order approximation functions and to implement h-adaptivity is fully exploited. A second-order moving-least squares approximation for both displacement and phase field is employed. Domain integration of the weak forms is evaluated by the quadratically consistent 3-point integration scheme. The refinement criterion using maximum residual strain energy history is proposed and the insertion of nodes is based on the background mesh. Numerical results show that the developed method is more efficient than the standard finite element method (3-node triangle element) due to the proposed h-adaptivity. In comparison with the standard EFG method, the proposed consistent EFG method significantly improves the computational efficiency and accuracy. The advantage of the quadratic approximation is also demonstrated. In addition, the feasibility of extending the proposed method to 3D is validated by the modeling of a twisting crack.

The successive node snapping scheme for an evolving branched curve in 2D and 3D

We introduce a novel method to construct conforming meshes for an evolving branched curve immersed in a given background mesh in 2D and 3D. The proposed method is built on the successive node snapping (SNS) scheme proposed by Wan and Shen (2016) and Wan et al. (2019) which is able to generate a conforming mesh for a simple evolving curve in 2D and 3D. The essence of the SNS scheme is to snap chosen nodes to the curve, and to simultaneously relax the nodes within the neighborhood of the curve. Following the curve, each portion of the curve’s vicinity is adjusted to obtain a mesh conforming to the entire curve. The core of the generalization of the SNS scheme for a branched curve is to assign each branch a matching node, from which we start to adjust the mesh for the branch via a procedure similar to that for a simple curve. The approach to determine the matching node is based on minimizing mesh distortion. The proposed approach inherits the main advantage of the SNS scheme, including unaltered nodal connectivities, small adjustment of nodes, no a priori conformity requirements on the background mesh, and permitting obtuse-angled triangular or tetrahedral background mesh as well as acute-angled ones. As all the operations on the mesh are local, the proposed method is especially suitable for evolving branched curve problems in which the curve updates a little bit at each time/ load step. Numerical examples in 2D and 3D are provided.

A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of elliptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in Rd, so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the background mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and condition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.