A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

Strain energy release rate calculation for a moving delamination front of arbitrary shape based on the virtual crack closure technique. Part II: Sensitivity study on modeling details

The interface element and VCCT process described in Part I of this two-part paper, developed to compute strain energy release rates of an arbitrary delamination front using non-orthogonal finite element meshes, are further investigated in this paper for robustness and ease of use in tracking delamination growth. Standard 3-D elements are used in conjunction with the interface elements. No special singularity elements are required. Stationary meshes that are independent of the shape of the delamination front can be used. Three cases having different initial delamination shapes are examined. The process is shown to be insensitive to the values used for the interfacial spring stiffness, the orientation of the interface element, or even the mesh pattern if the mesh has a reasonable degree of refinement. Therefore, the method can be used with ease and confidence in general-purpose delamination growth analysis for engineering applications.     

Adapting the Hypervolume Quality Indicator to Quantify Trade‐offs and Search Efficiency for Multiple Criteria Decision Making Using Pareto Fronts

When choosing a best solution based on simultaneously balancing multiple objectives, the Pareto front approach allows promising solutions across the spectrum of user preferences for the weightings of the objectives to be identified and compared quantitatively. The shape of the complete Pareto front provides useful information about the amount of trade‐off between the different criteria and how much compromise is needed from some criterion to improve the others. Visualizing the Pareto front in higher (3 or more) dimensions becomes difficult, so a numerical measure of this relationship helps capture the degree of trade‐off. The traditional hypervolume quality indicator based on subjective scaling for multiple criteria optimization method comparison provides an arbitrary value that lacks direct interpretability. This paper proposes an interpretable summary for quantifying the nature of the relationship between criteria with a standardized hypervolume under the Pareto front (HVUPF) for a flexible number of optimization criteria, and demonstrates how this single number summary can be used to evaluate and compare the efficiency of different search methods as well as tracking the search progress in populating the complete Pareto front. A new HVUPF growth plot is developed for quantifying the performance of a search method on completeness, efficiency, as well as variability associated with the use of random starts, and offers an effective approach for method assessment and comparison. Two new enhancements for the algorithm to populate the Pareto front are described and compared with the HVUPF growth plot. The methodology is illustrated with an optimal screening design example, where new Pareto search methods are proposed to improve computational efficiency, but is broadly applicable to other multiple criteria optimization problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

The mechanics of the interaction between a fluid and a soft interface undergoing large deformations appear in many places, such as in biological systems or industrial processes. We present an Eulerian approach that describes the mechanics of an interface and its interactions with a surrounding fluid via the so‐called Navier boundary condition. The interface is modeled as a curvilinear surface with arbitrary mechanical properties across which discontinuities in pressure and tangential fluid velocity can be accounted for using a modified version of the extended finite element method. The coupling between the interface and the fluid is enforced through the use of Lagrange multipliers. The tracking and evolution of the interface are then handled in a Lagrangian step with the grid‐based particle method. We show that this method is ideal to describe large membrane deformations and Navier boundary conditions on the interface with velocity/pressure discontinuities. The validity of the model is assessed by evaluating the numerical convergence for a axisymmetrical flow past a spherical capsule with various surface properties. We show the effect of slip length on the shear flow past a two‐dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

A BEM‐level set domain‐decomposition strategy for non‐linear and fragmented interfacial flows

A domain‐decomposition algorithm has been developed to handle two‐phase flows with large deformation, breaking and fragmentation of the interface. The strategy couples a boundary element method with a Navier–Stokes solver combined with a level‐set technique for the tracking of the interface. The former is used in the fluid region where the interface can be modelled as a smooth surface. In the rest of the domain the field solver is applied. This results in an efficient and accurate method. In this paper, the features of the used strategy are described and the challenges connected with the coupling are deeply discussed. The numerical investigation highlighted the importance of a proper rational study when CFD methods are considered. In the present case, a crucial aspect is represented by the domain‐composition step, that is when the information from one solver to the other have to be properly reconstructed and made consistent with the receiver sub‐domain. Copyright © 2006 John Wiley & Sons, Ltd.     

An enthalpy formulation based on an arbitrarily deforming mesh for solution of the Stefan problem

Traditionally schemes for dealing with the Stefan phase change problem are separated into fixed grif or front tracking (deforming grid) schemes. A standard fixed grid scheme is to use an enthalpy formulation and track the movement of the phase front via a liquid fraction variable. In this paper, an enthalpy formulation is applied on a continuously deforming finite element grid. This approach results in a general numerical scheme that incorporates both front tracking and fixed grid schemes. It is shown how on appropriate setting of the grid velocity a fixed or deforming grid solution can be generated from the general scheme. In addition an approximate front tracking scheme is developed which can produce accurate non-oscillatory predictions at a computational cost close to an efficient fixed grid scheme. The versatility of the general scheme and the approximate front tracking scheme are demonstrated on solution of a number of Stefan problems in both one and two dimensions.     

Evaluation of solid-liquid interface profile during continuous casting by a spline based formalism

A numerical framework has been applied which comprises of a cubic spline based collocation method to determine the solid-liquid interface profile (solidification front) during continuous casting process. The basis function chosen for the collocation algorithm to be employed in this formalism, is a cubic spline interpolation function. An iterative solution methodology has been developed to track the interface profile for copper strand of rectangular transverse section for different casting speeds. It is based on enthalpy conservation criteria at the solidification interface and the trend is found to be in good agreement with the available information in the literature although a point to point mapping of the profile is not practically realizable. The spline based collocation algorithm is found to be a reasonably efficient tool for solidification front tracking process, as a good spatial derivative approximation can be achieved incorporating simple modelling philosophy which is numerically robust and computationally cost effective.     

Uncertainty‐based robust aerodynamic optimization of rotor blades

The aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive backstepping motion control of induction motor drives

Abstract

In this paper, an adaptive backstepping controller is proposed for position tracking of a mechanical system driven by an induction motor. The mechanical system is a single link fixed on the shaft of the induction motor such as a single‐link robot. The backstepping methodology provides a simpler design procedure for an adaptive control scheme and provides a method to define the sliding surface if the robust slidingmode control is applied. Thus, the backstepping control can be easily extended to work as an adaptive sliding‐mode controller. The presented position control system is shown to be stable and robust to parameter variations and external disturbances. The effectiveness of the proposed controllers is demonstrated in experiments.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems

 This paper presents a two-dimensional model for identification of the phase change front in a continuous casting process. The transport phenomena encountered in the considered process are solved by Boundary Element Method (BEM). For the known set of external boundary conditions, the whole problem is solved in two subdomains separated by a phase change front whose position is updated during the iteration process. The solution scheme involves the application of a front tracking procedure based on using sensitivity coefficients to find the correct position of the phase change front modelled by Bezier splines. The main features of the developed algorithms were investigated by several numerical tests, the most important results of which are presented in this article. Received 6 November 2000     

A method for automated digital image-based tracking of delamination fronts in translucent glass fibre-laminated composite materials

A new automated digital image-based method for tracking both the position and shape of the crack front of delaminations in translucent glass fibre-laminated test specimens is proposed. The method is applicable to both fracture toughness tests and fatigue tests. The digital image-based algorithm identifies the newly formed crack area by the change in reflected light and then extracts both the crack front shape and the position. The method is used in a standard double cantilevered beam specimen test setup with the addition of light sources and a camera. The method does not rely on subjective test operator interpretations or calibrations. A reference pattern on the specimen is used to correctly map between the camera coordinate system and the specimen coordinate system, which makes the method insensitive to the camera position and camera or specimen movement during the test. Experimental test examples are provided, which shows that the whole crack front can be tracked with significantly higher accuracy than possible with existing methods.     

Infrared target tracking with kernel-based performance metric and eigenvalue-based similarity measure

An infrared target tracking framework is presented that consists of three main parts: mean shift tracking, its tracking performance evaluation, and position correction. The mean shift tracking algorithm, which is a widely used kernel-based method, has been developed for the initial tracking for its efficiency and effectiveness. A performance evaluation module is applied for the online evaluation of its tracking performance with a kernel- based metric to unify the tracking and performance metric within a kernel-based tracking framework. Then the tracking performance evaluation result is input into a controller in which a decision is made whether to trigger a position correction process. The position correction module employs a matching method with a new eigenvalue-based similarity measure computed from a local complexity degree weighted covariance matrix. Experimental results on real-life infrared image sequences are presented to demonstrate the efficacy of the proposed method.     

Two‐fluid model with interface sharpening

Two‐fluid models are applicable for simulations of all types of two‐phase flows ranging from separated flows with large characteristic interfacial length scales to highly dispersed flows with very small characteristic interfacial length scales. The main drawback of the two‐fluid model, when used for simulations of stratified flows, is the numerical diffusion of the interface. Stratified flows can be easily and more accurately solved with interface tracking methods; however, these methods are limited to the flows, that do not develop into dispersed types of flows. The present paper describes a new approach, where the advantage of the two‐fluid model is combined with the conservative level set method for interface tracking. The advection step of the volume fraction transport equation is followed by the interface sharpening, which preserves the thickness of the interface during the simulation. The proposed two‐fluid model with interface sharpening was found to be more accurate than the existing two‐fluid models. The mixed flow with both: stratified and dispersed flow, is simulated with the coupled model in this paper. In the coupled model, the dispersed two‐fluid model and two‐fluid model with interface sharpening are used locally, depending on the parameter which recognizes the flow regime. Copyright © 2010 John Wiley & Sons, Ltd.     

An element‐free Galerkin method for 3D crack propagation simulation under complicated stress conditions

This study develops an element‐free Galerkin method based on the moving least‐squares approximation to trace three‐dimensional crack propagation under complicated stress conditions. The crack surfaces are modelled by a collection of planar triangles that are added when cracks propagate. The visibility criterion is adopted to treat the screening effect of the cracks on the influenced domain of a Gaussian point. Cracks are assumed to propagate in the perpendicular planes at crack front points when the strain energy release rates reach the material fracture toughness. This method is unique in that it uses a nonlinear contact iterative algorithm to consider contributions of crack surface interaction to the global equilibrium equations, so that crack opening, sliding and closing under complicated stress states can be efficiently modelled. Two numerical examples of three‐dimensional quasi‐static crack propagation were modelled with satisfactory results. Copyright © 2012 John Wiley & Sons, Ltd.     

A Study of Diamond Growth Instability t High Temperature-High Pressure

In this paper, crystal growth instability of diamond was studied in a Fe-Ni-C system at high temperature-high pressure (HPHT). As any other crystal grown from solution, the flat or smooth growth interface of the diamond crystal is highly sensitive to growth conditions. The growth front interface should be of great importance to understand the diamond growth process. The presence of cellular growth interface by transmission electron microscopy indicated that there existed a narrow constitutional supercooling zone in front of the growth interface. Several parallel layers with cellular interface by TEM directly suggested that the diamond grows from the solution of carbon in the molten catalyst layer by layer, which is in accordance with the result obtained by scanning electron microscopy in this paper.Impurities are trapped by rapidly advancing growth layers during the diamond growth and they impose a great effect on the growth front stability. As the growth front interface approaches the impurity particle to a distance of about 10-5～10-7 cm, appreciable molecular forces begin to operate between them, and the impurity particle is trapped as the growth rate reaches a critical value. As a result, the driving force for crystallization under the impurity particles becomes smaller, the front buckles under the particle. An impurity naturally reduces the growth rate to a different extent.     

A Study of Diamond Growth Instability at High Temperature—High Pressure

In this paper, crystal growth instability of diamond was studied in a Fe-Ni-C system at high temperature-high pressure (HPHT). As any other crystal from solution, the flat or smooth growth interface of the diamond crystal is highly sensitive to growth conditions.The growth front interface should be of great importance to understand the diamond growth process.The presence of cellular growth interface by transmission electron microscopy indicated that there existed a narrow constitutional supercooling zone in front of the growth interface .Several parallel layers with cellular interface by TEM directly suggested that the diamond grows from the solution of carbon in the molten catalyst layer by layer, which is in accordance with the result obtained by scanning electron microscopy in this paper.Impurities are trapped by rapidly advancing growth layers during the diamond growth and they impose a great effect on the growth front stability.As the growth front interface approaches the impurity particle to a distance of about 10^-5-10^-7 cm, appreciable molecular forces begin to operate between them,and the impurity particle is trapped as the growth rate reaches a critical value.As a result,the driving force for crystallization under the impurity particles becomes smaller ,the front buckles under the particle.An impurity naturally reduces the growth rate to a different extent.     

A free-front tracking algorithm for a control-volume-based Hele–Shaw method

With free surface flow problems the position of the free front at a particular time step needs to be predicted. When simulating the flow of hot molten polymer in injection moulds this position is essential for the prediction of the position of the weld lines on the final product. Weld lines are important in that they indicate positions of diminished strength and spoil the aesthetics of the product. A code using a finite volume approach on the thin gap Hele–Shaw flow equations to simulate the filling stage of the injection moulding process has been developed which requires a suitable algorithm for the free-front tracking. The Volume of Fluid (VOF) method was implemented in this study for the tracking of the free front. The discretization and various formulations of the VOF equation are presented as well as the results obtained. The impact on the existing equations is also investigated as well as the development of a packing routine that is used to eliminate the diffusive nature of the VOF equation. © 1998 John Wiley & Sons, Ltd.     

Interface handling for three‐dimensional higher‐order XFEM‐computations in fluid–structure interaction

Three‐dimensional higher‐order eXtended finite element method (XFEM)‐computations still pose challenging computational geometry problems especially for moving interfaces. This paper provides a method for the localization of a higher‐order interface finite element (FE) mesh in an underlying three‐dimensional higher‐order FE mesh. Additionally, it demonstrates, how a subtetrahedralization of an intersected element can be obtained, which preserves the possibly curved interface and allows therefore exact numerical integration. The proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their ‘eXtended axis‐aligned bounding boxes’. The intersection method is applied to a highly reduced number of intersection candidates. The resulting linearized interface is used as input for an elementwise constrained Delaunay tetrahedralization, which computes an appropriate subdivision for each intersected element. The curved interface is recovered from the linearized interface in the last step. The output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element. Application of the interface algorithm currently concentrates on fluid–structure interaction problems on low‐order and higher‐order FE meshes, which may be composed of any arbitrary element types such as hexahedra, tetrahedra, wedges, etc. Nevertheless, other XFEM‐problems with explicitly given interfaces or discontinuities may be tackled in addition. Multiple structures and interfaces per intersected element can be handled without any additional difficulties. Several parallelization strategies exist depending on the desired domain decomposition approach. Numerical test cases including various geometrical exceptions demonstrate the accuracy, robustness and efficiency of the interface handling. Copyright © 2009 John Wiley & Sons, Ltd.