Dimensional Formulation Research Articles

Dendritic needle network modeling of the Columnar-to-Equiaxed Transition. Part II: three dimensional formulation, implementation and comparison with experiments

The dendritic needle network (DNN) model tracks the diffusion-controlled growth of branches in the hierarchically structured dendritic network, thereby bridging the well-separated scales traditionally simulated by phase-field and coarse-grained models. In particular, the DNN model relaxes the assumptions on the dendrite growth kinetics and grain structures commonly used in the coarse-grained models. In part I of this paper, we proposed a two-dimensional (2D) version of the DNN model and applied it to investigate the Columnar-to-Equiaxed Transition (CET) to clarify the influence of these assumptions on the prediction of the CET. In order to overcome the limitations inherent in a 2D model, part II presents here a fully three-dimensional (3D) version of the DNN model and its application to the CET. After validation of the 3D model, we perform simulations to study the solidification of Al-7 wt.% Si alloy and compare the results to the experimental measurements conducted on board of the International Space Station in the framework of the CET in SOLidification processing (CETSOL) project. The comparison shows that the present 3D DNN model is able to provide quantitative prediction of the position and the type of CET at experimental time and length-scales, without any adjustable parameters.

Dendritic needle network modeling of the Columnar-to-Equiaxed transition. Part I: two dimensional formulation and comparison with theory

The columnar-to-equiaxed transition is a technologically important phenomenon that controls the grain structure in the casting of alloys. Analytical and coarse-grained models, used to investigate this transition, rely on several assumptions concerning dendrite growth kinetics and grain structures. In the first part of this two part paper, we test these assumptions using a two-dimensional (2D) dendritic needle network model that describes both the transient growth dynamics of primary, secondary and higher order branches of the dendritic network within each grain and the solutal interactions between grains that can grow with arbitrary shapes. Our results provide novel insights into the columnar-to-equiaxed transition, distinguishing between abrupt and progressive transitions with different grain structures. Furthermore, they highlight the limitations of commonly used assumptions in analytical and coarse-grained numerical models of this transition. These results are extended to 3D in part II of the paper.

The GHSZ Argument: A Gedankenexperiment Requiring More Denken.

I reassess the gedankenexperiment of Greenberger, Horne, Shimony, and Zeilinger after twenty-five years, finding their influential claim to the discovery of an inconsistency inherent in high dimensional formulations of local realism to arise from a fundamental error of logic. They manage this by presuming contradictory premises: that a specific linear combination of four angles involved in their proposed parallel experiments on two pairs of electrons equals both and 0 at the same time. Ignoring this while presuming the contradictory implications of these two conditions, they introduce the contradiction themselves. The notation they use in their “derivation” is not sufficiently ornate to represent the entanglement in the double electron spin pair problem they design, confounding their error. The situation they propose actually motivates only an understanding of the full array of symmetries involved in their problem. In tandem with the error now recognised in the supposed defiance of Bell’s inequality by quantum probabilities, my reassessment of their work should motivate a reevaluation of the current consensus outlook regarding the principle of local realism and the proposition of hidden variables.

Convex Nonparametric Formulation for Identification of Gradient Flows

We develop a nonparametric identification method for nonlinear gradient-flow dynamics. In these systems, the vector field is the gradient field of a potential energy function. This fundamental fact about the dynamics is a structural prior knowledge and a constraint in the proposed identification method. While the nature of the identification problem is an estimation in the space of functions, we derive an equivalent finite dimensional formulation, which is a convex optimization in the form of a quadratic program. This provides scalability and the opportunity for utilizing recently developed large-scale optimization solvers. The central idea is representing the energy function as a difference of two convex functions and learning these convex functions jointly. Based on necessary and sufficient conditions for function convexity, the identification problem is formulated, and then, the existence, uniqueness and smoothness of the solution is addressed. We also illustrate and evaluate the method numerically with two demonstrative examples.

On an Improved State Parametrization for Soft Robots With Piecewise Constant Curvature and Its Use in Model Based Control

Piecewise constant curvature models have proven to be an useful tool for describing kinematics and dynamics of soft robots. However, in their three dimensional formulation they suffer from many issues limiting their range of applicability - as discontinuities and singularities - mainly concerning the straight configuration of the robot. In this work we analyze these flaws, and we show that they are not due to the piecewise constant curvature assumption itself, but that instead they are a byproduct of the commonly employed direction/angle of bending parametrization of the state. We therefore consider an alternative state representation which solves all the discussed issues, and we derive a model based controller based on it. Examples in simulation are provided to support and describe the theoretical results. When using the novel parametrization, the system is able to perform more complex tasks, with a strongly reduced computational burden, and without incurring in spikes and discontinuous behaviors.

Rigid slab foundation subjected to strike–slip faulting: mechanisms and insights

In this paper, the interaction of rigid slab foundations with a strike–slip fault is studied. A three-dimensional finite-element model of the soil–foundation system is developed, employing a thoroughly validated constitutive model that incorporates strain softening. The widely used rigid box type boundaries are compared to periodic boundaries, demonstrating their superiority. Allowing the soil to deform freely, periodic boundaries allow realistic simulation of fault rupture propagation through soil and minimisation of parasitic boundary effects. An approximate dimensional formulation is presented and verified by comparing the response of two self-similar systems. A parametric study is conducted, revealing two distinct kinematic mechanisms: a rotational one, where the foundation mainly rotates; and a translational one, where the foundation either displaces parallel to the fault trace or remains almost stationary. It is shown that the response is controlled by the presence of the foundation, acting as a kinematic constraint, which can be quantified by its polar moment of inertia, Iz. The rotational mechanism leads to positive bending moments of larger magnitude, whereas the translational leads to negative moments of larger magnitude. The effect of key dimensionless parameters is parametrically investigated, including the fault rupture location, the surcharge load, the interface friction coefficient and the foundation aspect ratio. Shorter foundations (parallel to the fault trace) tend to rotate, whereas longer ones tend to displace. Fault rupture locations close to the foundation sides lead to translation, whereas locations close to its centreline lead to rotation. The development of the translation mechanism is facilitated by the increase of surcharge load and interface friction coefficient.

Multivariate leverage effects and realized semicovariance GARCH models

We propose new asymmetric multivariate volatility models. The models exploit estimates of variances and covariances based on the signs of high-frequency returns, measures known as realized semivariances, semicovariances, and semicorrelations, to allow for more nuanced responses to positive and negative return shocks than threshold “leverage effect” terms traditionally used in the literature. Our empirical implementations of the new models, including extensions of widely-used bivariate GARCH specifications for a number of individual stocks and the aggregate market portfolio as well as larger dimensional dynamic conditional correlation type formulations for a cross-section of individual stocks, provide clear evidence of improved model fit and reveal new and interesting asymmetric joint dynamic dependencies.

ПРИМЕНЕНИЕ МЕТОДОЛОГИИ БЫСТРОГО АВТОМАТИЧЕСКОГО ДИФФЕРЕНЦИРОВАНИЯ К РЕШЕНИЮ ОБРАТНЫХ КОЭФФИЦИЕНТНЫХ ЗАДАЧ

The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general n -dimensional formulation. For the numerical solution of the problem an algorithm based on the modern fast automatic differentiation technique is proposed.

Dendritic Needle Network Modeling of the Columnar-to-Equiaxed Transition. Part II: Three Dimensional Formulation, Implementation and Comparison with Experiments

The dendritic needle network (DNN) model tracks the diffusion-controlled growth of branches in the hierarchically structured dendritic network, thereby bridging the well-separated scales traditionally simulated by phase-field and coarse-grained models. In particular, the DNN model relaxes the assumptions on the dendrite growth kinetics and grain structures commonly used in the coarse-grained models. In part I of this paper, we proposed a two-dimensional (2D) version of the DNN model and applied it to investigate the Columnar-to-Equiaxed Transition (CET) to clarify the influence of these assumptions on the prediction of the CET. In order to overcome the limitations inherent in a 2D model, part II presents here a fully three-dimensional (3D) version of the DNN model and its application to the CET. After validation of the 3D model, we perform simulations to study the solidification of Al-7wt.%Si alloy and compare the results to the experimental measurements conducted on board of the International Space Station (ISS)in the framework of the CET in SOLidification processing project. The comparison shows that the present 3D DNNmodel is able to provide quantitative prediction of the position and the type of CET at experimental time and length-scales, without any adjustable parameters.

Application of the Fast Automatic Differentiation Technique for Solving Inverse Coefficient Problems

Results obtained by the authors in solving inverse coefficient problems are overviewed. The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general $$n$$-dimensional formulation. For this general case, an analytical expression for the cost functional gradient is obtained. The features of solving the inverse problem and the difficulties encountered in the solution process are discussed.

Dendritic Needle Network Modeling of the Columnar-to-Equiaxed Transition. Part I: Two Dimensional Formulation and Comparison with Theory

Dendritic Needle Network Modeling of the Columnar-to-Equiaxed Transition. Part I: Two Dimensional Formulation and Comparison with Theory

Development of a fast EoS based compositional model for three-phase core flooding

Compositional simulators are of great importance to model core flow experiments and enhanced oil recovery processes. In the present study, a semi-implicit, three dimensional EoS-based formulation is developed for modeling and simulation of core-flooding process. The model treats three-phase flow in cylindrical coordinates with no need to use an iterative numerical method. Overall composition, pressure and water saturation are calculated implicitly for each grid block. Using this approach, a linear set of equations with constant coefficients would be solved at each time step. It is assumed that there is no mass transfer between hydrocarbon phases and water phase. In addition, a new technique has been suggested to calculate transmissibility coefficients in each direction, which is a major problem in compositional modeling. Each transmissibility coefficient is divided into two general parts which are treated differently. The first part is constant and the second part is time-dependent. This model yields a robust and fast calculation procedure, which is validated using Eclipse simulation results. Simulation of several miscible flooding processes were performed using Eclipse core-flood models in order to compare the results with the developed in-house model. Results are matched reasonably while the computing time is considerably decreased using the in-house model. It is shown that using second Stone’s relative permeability model and Advanced Peng–Robinson Equation of State, best results are achieved with an average absolute relative deviation of 7.7% from Eclipse results for ultimate recovery factor. In addition, sensitivity analysis has been performed on several input data including time-step size and block number. Results show that in a reasonable range of these parameters, convergence problems are not encountered. We have developed an in-house simulator based on our model, which can be a basis for other compositional simulation purposes. In addition to decreasing computing time, convergence problems encountered in most implicit techniques would be avoided as well.

Generalized Multiscale Finite Element method for multicontinua unsaturated flow problems in fractured porous media

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that is coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using Generalized Multiscale Finite Element Method (GMsFEM). In this method, we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction: (1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size.

Application of a mixed variational higher order plate theory towards understanding the deformation behavior of hybrid laminates

AbstractHybrid laminates containing an elastomer layer in addition to fiber reinforced polymer as well as metal layers have been found beneficial in compensating issues frequently found with traditional fiber metal laminates. Commonly used equivalent single‐layer shell and plate theories, however, are unable to account for the strong heterogeneous stiffness distribution of the constituents within the laminate. Furthermore, the transverse shear and normal deformations in the elastomer layer are expected to significantly influence the deformation of the neighboring laminae. An accurate depiction of these transverse stresses requires a multi‐layer shell theory as opposed to commonly used single‐layer formulations. Hence, a higher order mixed variational plate theory is applied in order to study and predict the mechanical behavior of such laminates, especially on a structural level where the computational effort forbids the use of a three dimensional continuum formulation.

Material model of pulp fibres

AbstractPulp fibres are a part of the paper fibre network and contribute to its non‐linear rate‐dependent behaviour [1]. An appropriate material model helps to quantify and understand these mechanisms, which are highly relevant to industry. The development of the constitutive equations is based on cyclic uniaxial tension tests [2]. For that first a one dimensional viscoelastic, viscoplastic model was designed to fit experimental data. This one dimensional model is then ported to a three dimensional formulation with the logarithmic strain framework [3]. The investigation shows that pulp fibres have a distinct plastic behaviour, which could be the cause of the non‐linear behaviour on the macroscopic level of the network.

A simple model for size effects in constrained shear

Simple shear of a ductile layer constrained between two rigid solids is analyzed using a conventional one dimensional strain gradient plasticity formulation. The aim of the analyses is to gain insight into the scaling relations and size effects seen in the experiments of Mu et al.(2014, 2016, 2017). Our analysis presumes that the deviation from previous predictions based on conventional strain gradient plasticity theory is not due to a defect in the theory but is due to a change in boundary condition that takes place when a sufficiently large plastic shear strain gradient develops at the layer boundaries. Predictions based on this presumption are qualitatively, and in some aspects, quantitatively consistent with a range of the observations.

A phase‐field model of brittle fracture for an isogeometric Reissner‐Mindlin shell formulation

AbstractOne of the biggest challenges in fracture modeling is the correct physical description of the fracture processes. Over the past decade, the well‐established phase‐field modeling framework has gained a lot of attention, due to its ability to predict the fracture phenomena adequately. It has shown very promising results for a three dimensional solid formulation and has been extended to an isogeometric Kirchhoff‐Love formulation in order to describe brittle fracture in plates and shells, see [1]. However, the Kirchhoff‐Love theory only describes shells in the thin regime. For thick shells, the deformation behavior and subsequently the fracture behavior additionally depends on the transverse shear strains which are not present in the Kirchhoff‐Love approach. In this work, the phase‐field fracture framework is extended to an isogeometric Reissner‐Mindlin shell formulation [2]. In this approach, the shell is described using the midsurface and a director vector field for the thickness direction. Subsequently, the phase‐field is also defined only on the midsurface. In order to distinguish the cracking behavior in tension and compression, as proposed in [1], the spectral decomposition of the strain for the tension‐compression split is done on the total strain, which varies through the thickness. The proposed method is tested for several numerical examples and compared to a three dimensional solid formulation and the Kirchhoff‐Love shell formulation in order to confirm its accuracy and efficiency.

Fluid topology optimization: Bio-inspired valves for aircraft engines

This work shows the introduction of three dimensional fluid topology optimization for the design of valves without moving parts that can be manufactured through 3D printing and used in aircraft engines. The design obtained is inherently safer than standard design with moving parts and it resembles closely to biological structures.A new three dimensional fluid topology optimization formulation to achieve such geometries is shown. The novelty of the method is linked to the applicability to realistic high Reynolds numbers, as in aircraft engines. This is the first time that such geometries are obtained for realistic operating conditions that are applicable to gas turbines.The optimal geometries are validated by using Computational Fluid Dynamics simulations.

Virtual element method for semilinear elliptic problems on polygonal meshes

In this paper, the virtual element method is employed to approximate semilinear elliptic problems over arbitrary polygonal meshes. The nonlinear load term is approximated by employing the orthogonal L2 projection. The finite dimensional formulation and its implementation are discussed in detail and optimal a priori error estimate in H1 norm is derived. Further, two numerical experiments are conducted in order to illustrate the performance of the proposed scheme and to numerically justify the theoretical convergence rate. It is observed that the proposed method yields optimal convergence rates in both L2 and H1 norm.

Acoustic waves scattered by elastic waveguides using a spectral approach with a 2.5D coupled boundary-finite element method

This work presents a two-and-a-half dimensional (2.5D) spectral formulation based on the finite element method (FEM) and the boundary element method (BEM) to study wave propagation in acoustic and elastic waveguides. The analysis involved superposing two dimensional (2D) problems with different longitudinal wavenumbers. A spectral finite element (SFEM) is proposed to represent waveguides in solids with arbitrary cross-section. Moreover, the BEM is extended to its spectral formulation (SBEM) to study unbounded fluid media and acoustic enclosures. Both approaches use Lagrange polynomials as element shape functions at the Legendre–Gauss–Lobatto (LGL) points. The fluid and solid subdomains are coupled by applying the appropriate boundary conditions at the limiting interface. The proposed method is verified by means of two benchmark problems: wave propagation in an unbounded acoustic medium and the scattering of waves by an elastic inclusion. The convergence and the computational effort are evaluated for different h−p strategies. Numerical results show good agreement with the reference solution. Finally, the proposed method is used to study the pressure field generated by an array of elastic fluid-filled scatterers immersed in an acoustic medium.