Treatment Of Essential Boundary Conditions Research Articles

General treatment of essential boundary conditions in reduced order models for non-linear problems

Inhomogeneous essential boundary conditions must be carefully treated in the formulation of Reduced Order Models (ROMs) for non-linear problems. In order to investigate this issue, two methods are analysed: one in which the boundary conditions are imposed in an strong way, and a second one in which a weak imposition of boundary conditions is made. The ideas presented in this work apply to the big realm of a posteriori ROMs. Nevertheless, an a posteriori hyper-reduction method is specifically considered in order to deal with the cost associated to the non-linearity of the problems. Applications to nonlinear transient heat conduction problems with temperature dependent thermophysical properties and time dependent essential boundary conditions are studied. However, the strategies introduced in this work are of general application.

A new formulation for imposing Dirichlet boundary conditions on non‐matching meshes

SummaryGenerating matching meshes for problems with complex boundaries is often an intricate process, and the use of non‐matching meshes appears as an appealing solution. Yet, enforcing boundary conditions on non‐matching meshes is not a straightforward process, especially when prescribing those of Dirichlet type. By combining a type of Generalized Finite Element Method (GFEM) with the Lagrange multiplier method, a new technique for the treatment of essential boundary conditions on non‐matching meshes is introduced in this manuscript. The new formulation yields a symmetric stiffness matrix and is straightforward to implement. As a result, the methodology makes possible the analysis of problems with the use of simple structured meshes, irrespective of the problem domain boundary. Through the solution of linear elastic problems, we show that the optimal rate of convergence is preserved for piecewise linear finite elements. Yet, the formulation is general and thus it can be extended to other elliptic boundary value problems. Copyright © 2015 John Wiley & Sons, Ltd.

New treatment of essential boundary conditions in EFG method by coupling with RPIM

One of major difficulties in the implementation of meshfree methods using the moving least square (MLS) approximation, such as element-free Galerkin method (EFG), is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpolation can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin(EFG)-radial point interpolation method (RPIM) is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.

Implementing of Displacement Boundary Conditions of Element-Free Galerkin Method and its Applications Used in Fracture Mechanics

The main draw back of the Moving Least Squares (MLS) approximate used in element free Galerkin method (EFGM) is its lack the property of the delta function. To alleviate difficulties in the treatment of essential boundary conditions in EFGM, the local transformation method and the boundary singular weight method, which are used in the reproducing kernel particle method, is combined with the element free Galerkin method. The computational method is given to analyze the stress intensity factors and the numerical simulation of crack propagation of two-dimentional problems of the elastic fracture analysis. The application examples reveal the effectiveness and feasibility of the present methods.

Transient heat conduction analysis using the MLPG method and modified precise time step integration method

The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples.

An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions

The NURBS-based isogeometric analysis offers a seamless integration between the CAD and subsequent finite element analysis and has been shown to be very effective for wide classes of problems due to its accurate geometric description. However, similar to the moving least square or reproducing kernel meshfree shape functions, the NURBS basis functions generally are not interpolatory functions. In this work it is shown that the direct imposition of the inhomogeneous essential boundary conditions to the NURBS control points may lead to significant errors with deteriorated rates of convergence. Consequently an improved formulation for NURBS-based isogeometric analysis is proposed. This is fulfilled by employing a transformation method that relates the control variables to the collocated nodal values at the essential boundary. By using open knot vectors, the resulting NURBS basis functions associated with the interior control points vanish at the boundary. Thus unlike the meshfree approximation, the NURBS control points can be clearly partitioned into boundary and interior groups. Therefore the transformation method can be only applied to the boundary control points via invoking collocation of physical values at a set of boundary points. The NURBS approximation enhanced with transformed physical boundary variables is kinematically admissible and the essential boundary conditions can be enforced straightforwardly like the finite element method. Several potential and elasticity problems evince that much higher solution accuracy with optimal convergence rates can be achieved by the present improved formulation compared with the method with direct imposition of essential boundary conditions to control variables.

An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for analyzing 2D and 3D transient heat transfer problems with mixed boundary conditions and complicated geometries. In the ES-PIM, shape functions are constructed using the polynomial PIM with the Delta function property for easy treatment of essential boundary conditions. A generalized smoothing technique is used to reconstruct the temperature gradient field within the edge-based smoothing domains. The generalized smoothed Galerkin weak form is then used to establish the discretized system equations. Our results show that the ES-PIM can provide more close-to-exact stiffness compared with the “overly-stiff” finite element method (FEM) and the “overly-soft” node-based smoothed point interpolation method (NS-PIM). Owing to this important property, the present ES-PIM provides more accurate solutions than standard FEM using the same mesh. As an example, a practical cooling system of the rapid direct plasma deposition dieless manufacturing is studied in detail using the present ES-PIM, and a set of “optional” processing parameters of fluid velocity and temperature are found.

Developments in the application of the generalized finite element method to thick shell problems

This paper develops and analyzes two techniques to extend the use of generalized finite element method techniques to structural shell problems. The first one is a procedure to define local domains for enrichment functions based on the use of pseudo-tangent planes. The second one is a procedure for imposing homogeneous essential boundary conditions and treatment of boundary layer problems by utilizing special functions. The main idea supporting the pseudo-tangent proposition is the separation of the geometric description, with its intrinsical distortions with respect to the physical domain, from the approximation space, which is defined in a locally undistorted domain. The treatment of essential boundary conditions allows an adequate enrichment in the boundary vicinity, preserving the completeness of the polynomials defining the basis functions. A set of numerical cases are tested in order to show the behavior of the proposed strategies, and a number of observations are drawn from the results, as follows. First, the technique of constructing the enrichment functions on a pseudo-tangent plane shows good results, even with strongly curved shell surfaces. With respect to the locking problem, the method behaves in a similar way as the classical hierarchical finite element methods, avoiding locking for appropriate levels of p-refinements. The procedure considered to impose essential boundary conditions in strong form appears to be more accurate than with the penalty or Lagrange multiplier methods. The inclusion of exponential modes for the treatment of boundary layers in shells provided extremely good results, even with integration elements much larger than the shell thickness.

Efficient Meshfree Computation with Fast Treatment of Essential Boundary Conditions for Industrial Applications

This paper presents a fast treatment of essential boundary conditions in three-dimensional (3D) meshfree computation for computational efficiency. Due to the loss of Kronecker delta properties in the meshfree shape functions, the imposition of essential boundary conditions is tedious, especially in 3D applications. The proposed boundary singular kernel (BSK) method introduces singularities to the kernel functions associated with the essential and kinematically constrained boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values, and consequently constraints can be imposed directly. In this work, the recovery of nodal value properties on essential boundary nodes is proved for general n-dimensional geometries. The extension of previously proposed two-dimensional BSK method to 3D formulation thus becomes straightforward, and essential boundary treatment consumes almost no additional cost to meshfree computation and makes the method affordable for industrial applications. The effectiveness of the proposed method is demonstrated in 3D metal forming examples.

An Algebraic Treatment of Essential Boundary Conditions in the Particle–Partition of Unity Method

This paper is concerned with the treatment of essential boundary conditions in meshfree methods. In particular, we focus on the particle-partition of unity method (PPUM). However, the proposed technique is applicable to any partition of unity-based approach. We present an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degrees of freedom that remain for the approximation of the PDE by simple linear algebra. Notably, our approach requires no restrictions on the distribution of the discretization points nor on the employed (local) approximation spaces. We attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces. Hence, the global splitting can be computed with (sub)linear complexity. Due to this direct splitting of the meshfree PPUM function space, we can implement a conforming local treatment of essential boundary data so that the realization of Dirichlet boundary values in the meshfree PPUM is straightforward. The presented approach yields an optimally convergent scheme, which is demonstrated by the presented numerical results.

A meshless method for stress-wave propagation in anisotropic and cracked media

Several transient wave propagation problems in anisotropic media and isotropic media with cracks are numerically analyzed by using a new numerical algorithm based on meshless local Petrov–Galerking (MLPG) method. In this algorithm, a novel modified Moving Least Squares (MLS) approximation is introduced to simplify the treatment of essential boundary conditions. By using a variant type of MLPG1 methods, the stabilized scheme of the discretized elasto-dynamic equations is obtained. Explicit central difference method with lumped mass matrix is used to solve the coupled ODEs to increase the efficiency of present algorithm. Visibility criterion is used to present the cracks, and the path-independent dynamic J′ integral is adopted to evaluate the dynamic stress intensity factors. The availability and accuracy of the present algorithm in solving dynamic problems in isotropic or anisotropic media with cracks are tested through the comparison with the results obtained by the LS-DYNA and the method of characteristic. Finally, the transient stress wave interacting with a slanted crack under an impact loading is investigated in detail, in which the ability of extracting the different stress-wave components in a complex acoustic field is also proved.

Numerical Simulation for Dynamic Crack Propagation by MLPG

In this paper, a new algorithm based on Meshless Local Petrov-Galerking (MLPG) method is presented for analyzing the crack dynamic propagation. A new modified Moving Least Squares approximation is proposed to simplify the treatment of essential boundary conditions. Explicit central difference with lumped mass matrix is adopted for the time integral. Visibility criterion with crack edge node adding technique and the maximum hoop stress criterion are used to describe the crack propagation and forecast the crack propagation direction. Based on this algorithm, three-point bend specimen for impact fracture test is investigated. Comparing the results with those obtained by the laser caustic method and high-speed photographs, the accuracy of the present algorithm is proved.

Numerical method for shape optimization using meshfree method

A numerical method for continuum-based shape design sensitivity analysis and optimization using the meshfree method is proposed. The reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method. Special features of the meshfree method from a sensitivity analysis viewpoint are discussed, including the treatment of essential boundary conditions, and the dependence of the shape function on the design variation. It is shown that the mesh distortion that exists in the finite element–based design approach is effectively resolved for large shape changing design problems through 2-D and 3-D numerical examples. The number of design iterations is reduced because of the accurate sensitivity information.

Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact

Two major difficulties are encountered in the meshfree solution of incompressible boundary value problems. The first is due to the employment of higher-order quadrature rules that leads to an over-constrained discrete system in incompressible problems. The second is associated with the treatment of essential boundary conditions and contact conditions owing to the loss of Kronecker delta properties in the meshfree shape functions. This paper discusses some recent enhancements in meshfree methods for incompressible boundary value problems, carries out numerical convergence analysis, and compares accuracy and efficiency improvement of these methods. Presented methods are a pressure projection method to remedy the over-constrained discrete system, and a mixed transformation method and a boundary singular kernel method for imposition of essential boundary conditions and contact constraints. Several linear and nonlinear problems were analyzed to demonstrate the effectiveness of the new approaches.

Explicit Reproducing Kernel Particle Methods for large deformation problems

The explicit Reproducing Kernel Particle Method (RKPM) is presented and applied to the simulations of large deformation problems. RKPM is a meshless method which does not need a mesh structure in its formulation. Because of this mesh-free property, RKPM is able to simulate large deformation problems without remeshing which is often required for the mesh-based methods such as the finite element method. The RKPM shape function and its derivatives are constructed by imposing the consistency conditions. An efficient treatment of essential boundary conditions is also proposed for explicit time integration. The Lagrangian method based on the reference configuration is employed for the RKPM simulation of large deformation problems. Several examples of non-linear elastic materials are solved to demonstrate the performance of the method. The numerical experiment for the problem of underwater bubble explosion is also performed using the explicit Lagrangian RKPM formulation. © 1998 John Wiley & Sons, Ltd.

An algorithm for permanent fluid-structure interaction in explicit transient dynamics

This paper presents the strategies and models adopted in the PLEXIS-3C computer code for the description of interactions of the permanent type between fluids and structures in fast transient dynamic problems. A careful analysis of geometrical issues and the use of a method based on Lagrange multipliers for the treatment of essential boundary conditions lead to a scheme which appears to be both general and robust, while remaining reasonably simple. The method is implemented via an algorithm able to deal automatically with situations of large geometrical complexity, such as those encountered with box-like structures—multiple interactions, structural bifurcations or even arbitrary structural intersections, submerged structural edges, etc.—in the presence of large structural displacements, rotations and deformations. The algorithm turns out to be efficient and dramatically reduces the amount of information that has to be supplied in order to describe a given interaction, compared with the algorithms used in previous PLEXIS-3C implementations. As a consequence, the time and effort needed for input preparation in large and complex, realistic simulations, is substantially reduced and the possibility for human errors in this phase is almost eliminated. Several examples show the application of the method to problems of practical interest.