An augmented Lagrangian method for discrete large-slip contact problems

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.     

An augmented finite element method for modeling arbitrary discontinuities in composite materials

An augmented finite element method (“A-FEM”) is presented that is a variant of the method of Hansbo and Hansbo (Comput Methods Appl Mech Eng, 193: 3523–3540, 2004), which can fully account for arbitrary discontinuities that traverse the interior of elements. Like the method of Hansbo and Hansbo, the A-FEM preserves elemental locality, because element augmentation is implemented within single elements and involves nodal information from the modified element only. The A-FEM offers the additional convenience that the augmentation is implemented via separable mathematical elements that employ standard finite element nodal interpolation only. Thus, the formulation is fully compatible with standard commercial finite element packages and can be incorporated as a user element without access to the source code. Because possible discontinuities include both elastic heterogeneity and cracks, the A-FEM is ideally suited to modeling damage evolution in structural or biological materials with complex morphology. Elements of a multi-scale approach to analyzing damage mechanisms in laminated or woven textile composites are used to validate the A-FEM and illustrate its possible uses. Key capabilities of the formulation include the use of meshes that need not conform to the surfaces of heterogeneities; the ability to apply the augmented element recursively, enabling modeling of multiple discontinuities arising on different, possibly intersecting surfaces within an element; and the ease with which cohesive zone models of nonlinear fracture can be incorporated.     

Improved decohesion modeling with the material point method for simulating crack evolution

A combined elastoplasticity and decohesion model is used with the material point method for the crack problem as described in the Sandia National Laboratories challenge. To predict the cracking path in a complex configuration with the least computational cost, the decohesion modeling is improved by making the failure mode adjustable and by replacing the critical normal and tangential decohesion strengths with the tensile and shear peak strengths, without performing discontinuous bifurcation analysis in each loading step after the onset of failure is identified. It is found that there is a transition between different failure modes along the cracking path, which depends on the stress distribution around the path due to the nonlocal nature of failure evolution. Based on the parametric study and available experimental data, the proposed model-based simulation procedure could be calibrated to predict the essential feature of the observed cracking response.     

Reduced order modeling of liquid sloshing in 3D tanks using boundary element method

This paper presents the application of reduced order modeling technique for investigation of liquid sloshing in three-dimensional tanks. The governing equations of sloshing are written using a boundary element formulation for incompressible potential flow. Then, the governing equations are reduced to a more efficient form that is represented only in terms of the velocity potential on the liquid free surface. This particular form is employed for eigen-analysis of fluid motion and the sloshing frequencies and mode shapes are determined. Then, the sloshing frequencies and the corresponding right- and left-eigenvectors are used along the modal analysis technique to find a reduced order model (ROM) for fluid motion. Using a rectangular and a cylindrical tank, the results of the ROM are verified in comparison with the analytical results. Then some example tanks are used to examine the ROM in comparison with the full boundary element model for determination of the free surface motion under the angular and lateral motion of the tank. The obtained results demonstrate the capability and accuracy of the reduced order modeling technique for analysis of sloshing using a few modes.     

A meshfree method for incompressible fluid dynamics problems

We show that meshfree variational methods may be used for the solution of incompressible fluid dynamics problems using the R‐function method (RFM). The proposed approach constructs an approximate solution that satisfies all prescribed boundary conditions exactly using approximate distance fields for portions of the boundary, transfinite interpolation, and computations on a non‐conforming spatial grid. We give detailed implementation of the method for two common formulations of the incompressible fluid dynamics problem: first using scalar stream function formulation and then using vector formulation of the Navier–Stokes problem with artificial compressibility approach. Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back‐facing step channel problem reveal strengths and weaknesses of the proposed meshfree method. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit potential method for incompressible flows

In this paper, we consider a novel numerical scheme for solving incompressible flows on collocated grids. The implicit potential method utilizes an implicit potential velocity obtained from a Helmholtz decomposition for the mass conservation and employs a modified form of Bernoulli's law for the coupling of the velocity–pressure corrections. It requires the solution only of the momentum equations, does not involve the solution of additional partial differential equations for the pressure, and is applied on a collocated grid. The accuracy of the method is tested through comparison with analytical, experimental, and numerical data from the literature, and its efficiency and robustness are evaluated by solving several benchmark problems such as flow around a circular cylinder and in curved square and circular ducts.Copyright © 2013 John Wiley & Sons, Ltd.     

Coupling lattice Boltzmann and material point method for fluid-solid interaction problems involving massive deformation

This paper presents a coupling lattice Boltzmann and material point method (LBMPM) for fluid-solid interaction problems involving massive deformation. The convected particle domain interpolation-based material point method is adopted to solve the structure responses due to the particularly advantage on dynamic massive deformation simulations and the lattice Boltzmann method is utilized for its reliability and simplicity to simulate the complex fluid flow. The coupling strategy for these two methods is based on the consistent conditions with respect to displacement, velocity, and force, respectively, on the interface between fluid and solid parts, including the unified interpolation bounce-back scheme for curved boundaries, the Galilean invariant momentum exchange method for hydrodynamic forces, the force imposing strategy particularly for massive deformation, and the refilling algorithm for moving boundaries. There is no remeshing operation needed in the proposed LBMPM for both solid and fluid parts even when solid massive deformation and fluid complex flow are considered. Three representative numerical examples are carried out and the simulation results demonstrate that the proposed LBMPM is capable of simulate complex bidirectional fluid-solid interaction processes with the superiority for the problems involving solid dynamic large deformation behaviors and complex fluid flow.     

An adaptive material point method coupled with a phase-field fracture model for brittle materials

In this study, a new adaptive method for crack propagation analysis is developed by using the material point method coupled with a phase-field fracture model for brittle materials. A background grid of material particles is adaptively refined based on the amount of material damage to resolve the length scale in the phase-field evolution equation. A division process of the material particles associated with the refined background cells is also performed to increase the resolution of solutions near the crack tip. The effectiveness and validity of the proposed method is assessed through several numerical examples for crack propagation in brittle materials.     

Temporal and null-space filter for the material point method

This paper presents algorithm improvements that reduce the numerical noise and increase the numerical stability of the material point method formulation. Because of the linear mapping required in each time step of the material point method algorithm, a possible mismatch between the number of material points and grid nodes leads to a loss of information. These null-space related errors may accumulate and affect the numerical solution. To remove the null-space errors, the presented algorithm utilizes a null-space filter. The null-space filter shown removes the null-space errors, resulting from the rank deficient mapping and the difference between the number of material points and the number of nodes. The presented algorithm enhancements also include the use of the explicit generalized-α integration method, which helps optimizing the numerical algorithmic dissipation. This paper demonstrates the performance of the improved algorithms in several numerical examples.     

Generalized interpolation material point method for coupled thermo-mechanical processes

In this paper, a generalized interpolation material point method (GIMP) for simulating coupled thermo-mechanical processes is developed based on the weak formulations of both conservation of momentum and conservation of energy. The coupling term between the thermal and mechanical field variables is investigated in detail. The convergence behavior of the GIMP is examined with respect to the number of background cells and the number of particles per cell, respectively. A multi-grid approach is proposed to improve the accuracy of the GIMP in dealing with the Dirichlet boundary conditions in thermal analyses. Several representative examples are presented to demonstrate and verify the proposed procedure as compared with analytical or other numerical solutions, including an example on coupled thermo-mechanical failure evolution. It is shown from the obtained results that the proposed procedure might provide a robust spatial-discretization tool for multi-physics simulations.     

Analytical modeling of liquid sloshing in a two-dimensional rectangular tank with a slat screen

Potential-flow theory is employed with linear free-surface conditions, multimodal method, and a screen-averaged pressure-drop condition to derive an analytical modal model describing the two-dimensional resonant liquid motions in a rectangular tank with a vertical slat-type screen in the tank middle. The tank is horizontally excited in a frequency range covering the two lowest natural sloshing frequencies. The model consists of a system of linear ordinary differential [modal] equations responsible for liquid sloshing in compartments, as well as a nonlinear ordinary differential equation describing the liquid flow between the compartments. New experimental model tests on steady-state wave elevations near the tank wall are reported for the solidity ratios 0.328 ?? Sn ?? 0.963 where Sn is the ratio between the solid area and the full area of the screen. The experiments generally support the applicability of the model. The discrepancy can be explained by the free-surface nonlinearity. The screen acts as a damping mechanism for low and intermediate solidity ratios, but it causes an increase in the lowest resonant sloshing frequency at higher solidity ratios as if the screen had been replaced by an unperforated wall.     

A new particle method for simulation of incompressible free surface flow problems

A new Lagrangian particle method called the consistent particle method (CPM), which solves the Navier–Stokes equations in a semi‐implicit time stepping scheme, is proposed in this paper. Instead of using kernel function as in some particle methods, partial differential operators are approximated in a way consistent with Taylor series expansion. A boundary particle recognition method is applied to help define the changing liquid domain. The incompressibility condition of free surface particles is enforced by an adjustment scheme. With these improvements, the CPM is shown to be robust and accurate in long time simulation of free surface flow particularly for smooth pressure solution. Two types of free surface flow problems are presented to verify the CPM, that is, two‐dimensional dam break and liquid sloshing in a rectangular tank. In the dam break example, the CPM solutions of pressure and wave elevation are in good agreement with published experimental results. In addition, an experimental study of water sloshing in tank on a shake table was conducted to verify the CPM solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Decomposition method for solving optimal material orientation problems

The strain energy density is considered as a measure of the stiffness/flexibility of the composite structure. A methodology for determining the stationary points of the strain energy density in anisotropic solids is developed. The methodology proposed is based on new problem formulation, derivation and analysis of optimality conditions, and decomposition method. The optimal material orientation problem is formulated in terms of strains. The optimality conditions derived cover different material symmetries, linear and also some non-linear material models. The complexity analysis of the optimality conditions has been performed. The proposed approach allows to divide the solution of the optimal material orientation problem into less complicated subtasks.     

New method for thermo-optical modeling in liquid crystals

To investigate the thermo-optical effect in liquid crystal(s) (LC)s the statistical methods of experiment planning are applied. The experiment was conducted in accordance with the technique of orthogonal central compositional planning. In the course of the experiment, physical parameters (scanning velocity, laser beam intensity, LC temperature) have been varied on five levels. The obtained mathematical model describes the dependence of light-scattering texture dimensions in the LC layer onphysical parameters of information by thermo-optical registration. Such a model allows us to determine the suitable function model to obtain the preset dimensions of the light-scattering texture in a LC layer.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

一种基于帘线/橡胶复合材料细观力学的精确建模方法

为研究适应性底座的受压膨胀力学特性,提出了一种基于纤维帘线/橡胶复合材料细观力学的精确建模方法.该方法建立在帘线与橡胶材料参数的准确取值这一基础上,其中橡胶材料采用Mooney-Rivilin本构模型进行描述,通过拉伸试验验证了本构模型的准确性,基于束帘线拉伸试验规律对帘线拉伸模量进行了修正.通过上述方法,对适应性橡胶底座受压膨胀过程进行了数值模拟与试验研究.结果表明:这一精确建模方法能够较好地模拟底座的受压膨胀特性,能够获取底座中帘线与橡胶材料的应力、应变的分布以及二者的变化规律.研究工作为适应性底座的进一步研究和实际应用提供了技术支撑.     

Meshfree point collocation method for elasticity and crack problems

A generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems. The derivatives are obtained from a local approximation so their evaluation is computationally very efficient. This meshfree point collocation method has other advantages: it does not require special treatment for essential boundary condition nor the time‐consuming integration of a weak form. Neither the connectivity of the mesh nor differentiability of the weight function is necessary. The accuracy of the solutions is exceptional and generally exceeds that of element‐free Galerkin method with linear basis. The performance and robustness are demonstrated by several numerical examples, including crack problems. Copyright © 2004 John Wiley & Sons, Ltd.