Essential Boundary Conditions Research Articles

Impact of patient-specific LVOT inflow profiles on aortic valve prosthesis and ascending aorta hemodynamics

Patient-specific models become increasingly important in cardiovascular research as they allow prediction of surgical procedures. While the left ventricular outflow profile is an essential boundary condition, it remains unknown before treatment takes place. To overcome this problem, hemodynamics after virtual valve replacement were calculated based on different inlet profiles at the left ventricular outflow tract: a generic plug profile and a profile derived from 4D-flow-MRI. Spatially averaged parameters within the aorta were not significantly altered using either profile. A generic profile might be sufficient for the prediction of hemodynamics, circumventing the problem of predicting change in patient-specific boundary conditions.

Bending Analysis of Moderately Thick Arbitrarily Shaped Plates with Point Supports Using Simple Hp Cloud Method

In this paper, a simple hp cloud method is developed for the static analysis of Mindlin plates with various shapes and different boundary conditions. A new application of the hp cloud method, “simple hp cloud method,” is developed based on Kronecker delta property, so that the essential boundary conditions can be imposed directly and as easily as finite element method. Contrary to the hp cloud method, the simple hp cloud method does not require utilizing Lagrange multipliers; hence, the cost of solving algebraic equations is decreased. Shepard functions are used for the partition of unity functions and complete polynomials of order less than or equal to 3 for the enrichment functions part. By using enough order of complete polynomials, the shear locking can be properly controlled. Numerical results are compared against some other solutions to illustrate the accuracy and efficiency of the present method. To show the applications of the simple hp cloud method, deflections and bending moments of quadrilateral, triangular and circular plates with different boundary conditions subjected to transversal distributed loading are considered. In addition, deflections of rectangular and skew plates with point supports are analyzed.

Impact of spatial accuracy on district energy simulations

An integrated approach using district energy simulations is essential to analyse energy efficient buildings and cities. Thereby, an accurate building representation in dimensions of space is considered as an essential boundary condition. Nonetheless, district energy simulations are often carried out using a limited set of archetype buildings. This paper analyses therefore the impact of a more accurate building representation in district energy models, for a given district. Three approaches to include building geometry with a different level of detail are extensively analysed through a comparison of geometrical properties, peak power, total energy use and overheating risk. The GIS-based approach is favoured for the design of district energy systems, as it enables a more accurate and automatic implementation of the spatial dimension of the dynamic energy simulation results.

The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations

In this paper, the complex variable reproducing kernel particle method (CVRKPM) for solving the bending problems of isotropic thin plates on elastic foundations is presented. In CVRKPM, one-dimensional basis function is used to obtain the shape function of a two-dimensional problem. CVRKPM is used to form the approximation function of the deflection of the thin plates resting on elastic foundation, the Galerkin weak form of thin plates on elastic foundation is employed to obtain the discretized system equations, the penalty method is used to apply the essential boundary conditions, and Winkler and Pasternak foundation models are used to consider the interface pressure between the plate and the foundation. Then the corresponding formulae of CVRKPM for thin plates on elastic foundations are presented in detail. Several numerical examples are given to discuss the efficiency and accuracy of CVRKPM in this paper, and the corresponding advantages of the present method are shown.

Nonlinear and Transient Heat Transfer in the Fin by a Truly Meshless Method

In this article, Meshless Local Petrov- Galerkin (MLPG) method is used to solve the nonlinear and transient one- dimensional heat transfer equation of a fin with the power- law temperature- dependent heat transfer coefficient. Moving least square approximants are used to approximate the unknown function of temperature T(x) with Th (x). These approximants are constructed by using a linear basis, a weight function and a set of non- constant coefficients. Essential boundary conditions are enforced by direct method of interpolation and Penalty Method (PM) respectively. Temperature variation along the fin length over the different time range till the attainment of steady state has been demonstrated for the convective and insulated tip conditions.

An Improved Interpolating Complex Variable Meshless Method for Bending Problem of Kirchhoff Plates

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

A Fast Complex Variable Element-Free Galerkin Method for Three-Dimensional Wave Propagation Problems

In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin (ICVEFG) method, we present a fast ICVEFG method for three-dimensional wave propagation problems. Using the dimension splitting method, the equations of three-dimensional wave propagation problems are translated into a series of two-dimensional ones in another one-dimensional direction. The new Galerkin weak form of the dimension splitting method for three-dimensional wave propagation problems is obtained. The improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions, finite difference method is used in the one-dimensional direction, then the formulae of the ICVEFG method for three-dimensional wave propagation problems are obtained. The convergence and the corresponding parameters in the ICVEFG method are discussed. Some numerical examples are given to show that the new method has higher computational precision, and can improve the computational efficiency of the conventional meshless methods for three-dimensional problems greatly.

A hybrid improved complex variable element-free Galerkin method for three-dimensional potential problems

Combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional potential problems. Using the dimension splitting method, a three-dimensional potential problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method for each two-dimensional problem, the improved complex variable moving least-square (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional potential problem is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional potential problems is presented. Four numerical examples are given to show that the new method has higher computational efficiency.

Imposition of essential boundary conditions in the material point method

SummaryThere is increasing interest in the material point method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming; however, some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper, we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, for the MPM, expanding its capabilities to model boundaries of any inclination. To the authors' knowledge, this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and nonzero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that (1) the boundaries are independent of the background mesh, (2) artificially stiff regions of material points are avoided, and (3) the method does not rely on mirroring of the problem domain to impose symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM.

Limit state analysis of reinforced concrete slabs using an integrated radial basis function based mesh-free method

Limit analysis, a general and powerful method to predict the collapse load of reinforced concrete slabs, combines the limit theorems of classical plasticity with numerical methods, and results in an optimization problem. The number of variables in the corresponding optimization problem depends on the number of degrees of freedom used per node. In this paper, a rotation-free meshfree method based on integrated radial basis functions (iRBF) is developed. There is no rotational degree of freedom involved in the approximation, and hence the size of the resultant optimization problem is kept to a minimum. Moreover, the iRBF based shape functions hold the Kronecker delta property, ensuring that there is no need for any special treatment when enforcing essential boundary conditions. The obtained discrete kinematic formulation is handled using available highly efficient solvers. Various Nielsen’s reinforced concrete slabs of arbitrary geometries are examined, showing that the proposed numerical procedure is capable of providing accurate collapse load multipliers and automatically identifying yield-patterns in terms of plastic dissipation distribution.

Interpolating element-free Galerkin method for the regularized long wave equation and its error analysis

Combining the simplified interpolating moving least-squares (IMLS) method with the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method is presented for the numerical study of the regularized long wave (RLW) equation. The trial function in the IMLS method has one less unknown coefficient than in the traditional MLS approximation, and the shape functions satisfy the interpolating property at nodes. Then the IEFG method can apply the essential boundary conditions naturally and directly, and has high computational accuracy. And then this paper studies the error estimates of the IEFG method for RLW equation. The error estimate of the semi-discrete IEFG method for the RLW equation on the influence domain radius is first studied, and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied. The theoretical results show that the IEFG method for RLW equation has good convergence rate. For the purpose of demonstration, three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper.

Steady heat transfer analysis of orthotropic structure based on Element-Free Galerkin method

The mathematical model of steady heat transfer analysis for orthotropic structure is established based on Element-Free Galerkin (EFG) method and the essential boundary conditions are enforced using penalty method. The reliability of heat transfer model and MATLAB programs have been verified through three engineering examples in practice. The effects of nodes distribution pattern, scaling factors, penalty factors and weight functions on the EFG temperature results of different orthotropic structures are evaluated. A comparison of the CPU computing time between EFG and FEM is also studied. By comparison with finite element temperature and temperature gradient, it is revealed that the temperature based on EFG has higher calculation precision for steady heat transfer problem, and the temperature is not affected by nodes distribution pattern. The rational values of scaling factors and penalty factors are provided respectively. In addition, the computational relative error increases with the decrease of the orthotropic factor during various weight functions, and the cubic spline and parabola weight functions are the preferred choices in the heat transfer analysis of the orthotropic structure.

HIS-TORIC: extending core ICRF wave simulation to include realistic SOL plasmas

This paper presents a novel approach to incorporating an arbitrarily shaped edge scrape-off-layer (SOL) plasma and an ion cyclotron range of frequency (ICRF) antenna structure into existing core ICRF wave simulation models. We partition the entire computation domain into two sub-domains: a core and an edge region. Simulations in each domain are performed separately with appropriate numerical solvers. For the core, the TORIC ICRF solver (Brambilla 1999 Plasma Phys. Control. Fusion 41 1) was modified to impose an essential (Dirichlet) boundary condition at its interface with the edge domain. In the edge, a finite element method is used to solve a cold collisional plasma model. The domains are then joined together using the continuity boundary condition for the tangential electric and magnetic fields at their interfaces (Hybrid Integration of SOL to TORIC: HIS-TORIC). The model developed here was tested using an ICRH H minority heating scenario on the Alcator C-Mod tokamak (Hutchinson et al 1994 Phys. Plasmas 1 1511). The simulated pattern of core wave propagation agrees well with a standard TORIC simulation. This approach opens the possibility of using a realistic diverted SOL plasma and a complicated 3D RF antenna together with a rigorous hot core plasma model, while requiring only minimal modification to existing RF codes.

The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

Based on the improved complex variable moving least-squares (ICVMLS) approximation, the improved complex variable element-free Galerkin (ICVEFG) method for the bending problem of Kirchhoff plate is presented. Compared with the moving least-squares (MLS) approximation, in the ICVMLS approximation, the approximation function of two-dimensional problems can be obtained with one-dimensional basis function, then the computational efficiency of the shape functions is higher. Compared with the meshless methods based on the MLS approximation, under the same node distributions, the ones using the ICVMLS approximation can obtain the solutions with higher computational accuracy; and under the similar computational accuracy, the ones using the ICVMLS approximation have higher computational efficiency. The ICVMLS approximation is used to form the approximation function of the deflection of a Kirchhoff plate, the Galerkin weak form of the bending problem of Kirchhoff plates is adopted to obtain the discretized system equations, and the penalty method is employed to enforce the essential boundary conditions, then the corresponding formulae of the ICVEFG method for the bending problem of Kirchhoff plates are presented. By computing and analyzing four typical examples, it is shown that the ICVEFG method of Kirchhoff plates in this paper is efficient. And the computational precision of the numerical solutions is analyzed to select the basis function, weight function, scaling factor, node distribution and penalty factor in the ICVEFG method. Numerical examples show that the method in this paper has better convergence and higher accuracy. When the quadratic polynomial basis function and the cubic or quartic spline weight function are used, and d m a x = 4.2−4.4 , the ICVEFG method of Kirchhoff plates in this paper can obtain the solutions with high computational accuracy. And more nodes are distributed in the problem domain, higher computational accuracy of the solution will obtained, which shows that the method in this paper has better convergence.

A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems

By combining the interpolating moving least squares (IMLS) method with the variational multiscale method, a variational multiscale interpolating element-free Galerkin (VMIEFG) method is presented in this paper for the numerical solutions of convection-diffusion and Stokes problems. In the VMIEFG, the IMLS is used to construct shape functions based on using shifted and scaled polynomial bases. Compared with the variational multiscale element-free Galerkin (VMEFG) method, the VMIEFG method can directly apply the essential boundary conditions. The VMIEFG method is an effective meshless method, especially, for convection-dominated problems. Numerical examples show that the VMIEFG method avoids the oscillation in the element-free Galerkin (EFG) method, and the computational precision of the VMIEFG method is higher than that of the EFG, the VMEFG and the finite element methods.

Complex Variable Meshless Manifold Method for Transient Heat Conduction Problems

On the basis of the complex variable meshless manifold method (CVMMM), the time domain is discretized using the traditional two-point difference method, and the complex variable moving least-squares (CVMLS) method is used in the spatial domain. According to the equivalent integral weak form of the governing equation, the CVMMM for transient heat conduction problem is established, and the corresponding discreted equations are derived by introducing the penalty function method to the essential boundary conditions. The numerical examples show that compared with the meshless manifold method (MMM), the CVMMM for transient heat conduction problem has the advantages of high precision and less calculation. This proposed method is a new numerical calculation method which has great development potential.

Naturally stabilized nodal integration meshfree formulations for analysis of laminated composite and sandwich plates

This article addresses a naturally stabilized nodal integration (NSNI) meshfree formulation for static, free vibration and buckling analyses of laminated composite and sandwich plates based on the higher order shear deformation theory (HSDT). The crucial idea is to directly compute integrations at nodes similar to the direct nodal integration (DNI) but the instability existing in DNI is avoided and the computational cost is significantly reduced when compared to traditional high-order Gauss quadrature schemes. Due to the displacements, strains and stresses directly computed at nodes, the computing time of the present approach is equivalent with the DNI. Being different from the stabilized conforming nodal integration technique using a divergence theorem to evaluate gradient strains by the surface-to-boundary integration, NSNI is a naturally implicit gradient expansion. Variational consistency in Galerkin weak form is explicitly described. Thanks to the satisfied Kronecker delta function property, the moving Kriging integration (MKI) shape function is used to impose directly essential boundary conditions as same as the finite element method (FEM). A variety of numerical examples with various complex geometries, aspect ratios, stiffness ratios and boundary conditions are studied. Numerical results show the effectiveness of the present method in comparison with some existing ones in the literature.

Buckling of moderately thick arbitrarily shaped plates with intermediate point supports using a simple hp-cloud method

This paper presents the simple hp-cloud meshless method for the stability analysis of moderately thick plates with various shapes subjected to uniaxial and biaxial in-plane compressive and pure shear loads. To allow the effect of transverse shear deformation on the critical buckling load of the plate, the Mindlin's plate theory is employed. Simple hp-cloud method is applied for constructing the cloud shape functions and for discretization of the domain. Shepard functions are also used for the partition of unity and complete polynomials are utilized for the enrichment functions part. The simple hp-cloud method has Kronecker delta property, so contrary to most of meshless methods the essential boundary conditions can be imposed directly. Constructing the stiffness and geometry matrices leads to an eigenvalue problem that should be solved to determine the critical buckling load of the plate. Numerical results are verified against the results reported elsewhere to illustrate the accuracy and effectiveness of the present method. To show the applications of simple hp-cloud method in the buckling analysis of moderately thick plates, local buckling coefficients of various shapes of plates, rectangular, skew, trapezoidal, triangular, circular, semi-circular, hexagonal and general shape with different boundary conditions are determined. Also, local buckling coefficients of plates with point supports and intermediate support are calculated.

Application of Meshless Local Petrov–Galerkin Method in Inhomogeneous Electromagnetic Field

It is difficult to deal with the essential boundary conditions and the discontinuous medium boundary conditions for element-free galerkin method (EFG), because of the particularity of approximation function. In this paper, we use the visual principle to deal with discontinuous media boundary conditions equivalently. We apply discontinuous boundaries and essential boundary conditions to this method in a simpler way. This approach simplifies the processing of complex boundaries. In the article, the method is used to deal with the inhomogeneous medium model. The comparison between the exact solution and the numerical solution shows that the method is feasible and effective.

Closed form solution for a nonlocal strain gradient rod in tension

A size-dependent integral elasticity model is developed for a small-scaled rod in tension based on the nonlocal strain gradient theory. The integral rod model contains a nonlocal parameter and a material length scale parameter to incorporate the scaling effects of nonlocal stress and microstructure-dependent strain gradient. In comparison to size-dependent differential models, the developed integral rod model is both self-consistent and well-posed. The governing equations and boundary conditions for the nonlocal strain gradient rod in tension are derived by employing the principle of virtual work. In addition to the classical natural and essential boundary conditions, non-classical natural and essential boundary conditions are present for the integral rod model. The closed-form solutions for predicting the displacement and reduced Young’s modulus are derived for four types of boundary conditions. It is shown explicitly that the integral rod model can exert stiffness-softening and stiffness-hardening effects by considering various values of size-dependent parameters and boundary conditions. It is found that, the developed rods with four different boundary conditions can predict the scaling effects of the Young’s modulus of single-walled carbon nanotube, and the scaling effects are more sensitive to the size-dependent parameters (the material length scale parameter and the nonlocal parameter) in comparison with the non-classical boundary conditions.