etd@IISc Collection:
http://etd.iisc.ernet.in/2005/27
2018-04-19T00:40:10ZWing in Ground Effect
http://etd.iisc.ernet.in/2005/2848
Title: Wing in Ground Effect
Authors: Mondal, Partha
Abstract: The thesis presents a two pronged approach for predicting aerodynamics of air- foils/wings in the vicinity of the ground. The ﬁrst approach is eﬀectively a model for ground eﬀect studies, employing an inexpensive Discrete Vortex Method for the 2D pre- dictions and the well known Numerical lifting line theory for the 3D predictions. The second one pertains to the dynamic ground eﬀect analysis which employs the state of the art moving mesh methodology based time accurate CFD. In that sense, the thesis deals with two ends of spectrum in the ground eﬀect analysis; one, a model to be used in the concept design phase and the other an advanced CFD tool for analysis.
The proposed model for ground eﬀect studies is based on the well known Discrete Vortex Method (DVM). An important aspect of this method is that it employs what is referred to as the Generalized Kutta Joukowski Theorem (GKJ), meant for interaction problems with multiple vortices, for predicting the lift (and drag) within a potential ﬂow framework. After ascertaining the correctness of using the GKJ theorem for lift prediction for airfoils in ground eﬀect, a modiﬁed DVM is presented as a model for ground eﬀect predictions. As per this model, knowing the free stream lift and drag (either from an ex- periment or from a RANS computation) the aerodynamics of the section in ground eﬀect can be predicted. The model is eﬀectively built by constraining the DVM to produce the reference lift/drag in the free stream. The accuracy of the model, particularly for the more relevant high lift sections used during take-oﬀ and landing, is systematically estab- lished for a number of test cases. Knowing the sectional ground eﬀect, the extension to 3D analysis is very simple and this is achieved through the well known Numerical Lifting Line theory. The eﬃcacy of the proposed method for the 3D applications is demonstrated using a high lift wing in ground eﬀect. It is worth noting that the proposed model predicts the lift and drag very accurately, practically at no computational cost as compared to modern RANS based CFD tools requiring over 40 or 50 million volumes at a high computational cost and intense human intervention for generating the grids for every ground clearance.
The other aspect of the thesis pertains to what is referred to as the Dynamic Ground Eﬀect. Normally the CFD computations mimic the ground eﬀect experiments in simulat- ing the ground eﬀect. These simulations do not maintain geometric similarity with the actual landing or take-oﬀ sequence of the aircrafts and this can only be achieved when the simulations are dynamic. Dynamics is also important in case of combat aircrafts (particularly their naval versions) with an aggressive landing and take-oﬀ. The dynamic ground eﬀect simulations also provides a framework for simulating varied gust conditions. This dynamic simulation of the ground eﬀect is accomplished using a novel sinking grid methodology, which allows the grids to sink in the ground as the aircraft approaches the ground along the glide path. These simulations make use of the state of the art, time accurate moving grid methods and therefore can be computationally expensive. Never- theless, the utility of such computations in terms of their ability to produce continuous data has been highlighted in the thesis. In that sense, these dynamic computations will be cheaper as compared to the static simulations to produce data at the same level of resolution.2017-11-30T18:30:00ZWeighted Least Squares Kinetic Upwind Method Using Eigendirections (WLSKUM-ED)
http://etd.iisc.ernet.in/2005/538
Title: Weighted Least Squares Kinetic Upwind Method Using Eigendirections (WLSKUM-ED)
Authors: Arora, Konark
Abstract: Least Squares Kinetic Upwind Method (LSKUM), a grid free method based on kinetic
schemes has been gaining popularity over the conventional CFD methods for computation
of inviscid and viscous compressible ﬂows past complex conﬁgurations. The main reason
for the growth of popularity of this method is its ability to work on any point distribution. The grid free methods do not require the grid for ﬂow simulation, which is an essential requirement for all other conventional CFD methods. However, they do require point distribution or a cloud of points.
Point generation is relatively simple and less time consuming to generate as compared
to grid generation. There are various methods for point generation like an advancing front method, a quadtree based point generation method, a structured grid generator, an unstructured grid generator or a combination of above, etc. One of the easiest ways of point generation around complex geometries is to overlap the simple point distributions generated around individual constituent parts of the complex geometry. The least squares grid free method has been successfully used to solve a large number of ﬂow problems over the years. However, it has been observed that some problems are still encountered while
using this method on point distributions around complex conﬁgurations. Close analysis
of the problems have revealed that bad connectivity of the nodes is the cause and this leads to bad connectivity related code divergence.
The least squares (LS) grid free method called LSKUM involves discretization of
the spatial derivatives using the least squares approach. The formulae for the spatial derivatives are obtained by minimizing the sum of the squares of the error, leading to a system of linear algebraic equations whose solution gives us the formulae for the spatial derivatives. The least squares matrix A for 1-D and 2-D cases respectively is given by
(Refer PDF File for equation)
The 1-D LS formula for the spatial derivatives is always well behaved in the sense that ∑∆xi2 can never become zero. In case of 2-D problems can arise. It is observed that the elements of the Ls matrix A are functions of the coordinate differentials of the nodes in the connectivity. The bad connectivity of a node thus can have an adverse effect on the nature of the LS matrices. There are various types of bad connectivities for a node like insufficient number of nodes in the connectivity, highly anisotropic distribution of nodes in the connectivity stencil, the nodes falling nearly on a line (or a plane in 3-D), etc. In case of multidimensions, the case of all nodes in a line will make the matrix A singular thereby making its inversion impossible. Also, an anisotropic distribution of nodes in
the connectivity can make the matrix A highly illconditioned thus leading to either loss in accuracy or code divergence. To overcome this problem, the approach followed so far is to modify the connectivity by including more neighbours in the connectivity of the node. In this thesis, we have followed a diﬀerent approach of using weights to alter the nature of the LS matrix A.
(Refer PDF File for equation)
The weighted LS formulae for the spatial derivatives in 1-D and 2-D respectively are
are all positive. So we ask a question : Can we reduce the multidimensional LS formula for the derivatives to the 1-D type formula and make use of the advantages of 1-D type
formula in multidimensions?
Taking a closer look at the LS matrices, we observe that these are real and symmetric
matrices with real eigenvalues and a real and distinct set of eigenvectors. The eigenvectors of these matrices are orthogonal. Along the eigendirections, the corresponding LS formulae reduce to the 1-D type formulae. But a problem now arises in combining the eigendirections along with upwinding. Upwinding, which in LS is done by stencil splitting, is essential to provide stability to the numerical scheme. It involves choosing a direction for enforcing upwinding. The stencil is split along the chosen direction. But it is not necessary that the chosen direction is along one of the eigendirections of the split stencil. Thus in general we will not be able to use the 1-D type formulae along the chosen direction. This diﬃculty has been overcome by the use of weights leading to WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using Eigendirections). In WLSKUM-ED weights are suitably chosen so that a chosen direction becomes an eigendirection of A(w). As a result, the multi-dimensional LS formulae reduce to 1-D type formulae along the eigendirections. All the advantages of the 1-D LS formuale can thus be made use of even in multi-dimensions.
A very simple and novel way to calculate the positive weights, utilizing the coordinate
diﬀerentials of the neighbouring nodes in the connectivity in 2-D and 3-D, has been
developed for the purpose. This method is based on the fact that the summations
of the coordinate differentials are of diﬀerent signs (+ or -) in different quadrants or octants of the split stencil. It is shown that choice of suitable weights is equivalent to a suitable decomposition of vector space. The weights chosen either fully diagonalize the least squares matrix ie. decomposing the 3D vector space R3 as R3 = e1 + e2 + e3, where e1, e2and e3are the eigenvectors of A (w) or the weights make the chosen direction the eigendirection ie. decomposing the 3D vector space R3 as R3 = e1 + ( 2-D vector space R2). The positive weights not only prevent the denominator of the 1-D type LS formulae from going to zero, but also preserve the LED property of the least squares method. The WLSKUM-ED has been successfully applied to a large number
of 2-D and 3-D test cases in various ﬂow regimes for a variety of point distributions
ranging from a simple cloud generated from a structured grid generator (shock reﬂection
problem in 2-D and the supersonic ﬂow past hemisphere in 3-D) to the multiple chimera
clouds generated from multiple overlapping meshes (BI-NACA test case in 2-D and
FAME cloud for M165 conﬁguration in 3-D) thus demonstrating the robustness of the
WLSKUM-ED solver. It must be noted that the second order acccurate computations
using this method have been performed without the use of the limiters in all the ﬂow regimes. No spurious oscillations and wiggles in the captured shocks have been observed, indicating the preservation of the LED property of the method even for 2ndorder accurate computations.
The convergence acceleration of the WLSKUM-ED code has been achieved by the use
of LUSGS method. The use of 1-D type formulae has simplified the application of LUSGS method in the grid-free framework. The advantage of the LUSGS method is that the
evaluation and storage of the jacobian matrices can be eliminated by approximating the split flux jacobians in the implicit operator itself. Numerical results reveal the attainment of a speed up of four by using the LUSGS method as compared to the explicit time marching method.
The 2-D WLSKUM-ED code has also been used to perform the internal ﬂow computations. The internal ﬂows are the ﬂows which are confined within the boundaries. The inflow and the outflow boundaries have a significant effect on these ﬂows. The
accurate treatment of these boundary conditions is essential particularly if the ﬂow condition at the outflow boundary is subsonic or transonic. The Kinetic Periodic Boundary Condition (KPBC) which has been developed to enable the single-passage (SP) ﬂow computations to be performed in place of the multi-passage (MP) ﬂow computations,
utilizes the moment method strategy. The state update formula for the points at the periodic boundaries is identical to the state update formula for the interior points and can be easily extended to second order accuracy like the interior points. Numerical results have shown the successful reproduction of the MP ﬂow computation results using the SP ﬂow computations by the use of KPBC. The inflow and the outflow boundary conditions at the respective boundaries have been enforced by the use of Kinetic Outer Boundary Condition (KOBC). These boundary conditions have been validated by performing the ﬂow computations for the 3rdtest case of the 4thstandard blade conﬁguration of the turbine blade. The numerical results show a good comparison with the experimental results.2009-06-24T07:07:29ZWave Transmission Characteristics in Honeycomb Sandwich Structures using the Spectral Finite Element Method
http://etd.iisc.ernet.in/2005/2901
Title: Wave Transmission Characteristics in Honeycomb Sandwich Structures using the Spectral Finite Element Method
Authors: Murthy, MVVS
Abstract: Wave propagation is a phenomenon resulting from high transient loadings where the duration of the load is in µ seconds range. In aerospace and space craft industries it is important to gain knowledge about the high frequency characteristics as it aids in structural health monitoring, wave transmission/attenuation for vibration and noise level reduction.
The wave propagation problem can be approached by the conventional Finite Element Method(FEM); but at higher frequencies, the wavelengths being small, the size of the finite element is reduced to capture the response behavior accurately and thus increasing the number of equations to be solved, leading to high computational costs. On the other hand such problems are handled in the frequency domain using Fourier transforms and one such method is the Spectral Finite Element Method(SFEM). This method is introduced first by Doyle ,for isotropic case and later popularized in developing specific purpose elements for structural diagnostics for inhomogeneous materials, by Gopalakrishnan. The general approach in this method is that the partial differential wave equations are reduced to a set of ordinary differential equations(ODEs) by transforming these equations to another space(transformed domain, say Fourier domain). The reduced ODEs are usually solved exactly, the solution of which gives the dynamic shape functions. The interpolating functions used here are exact solution of the governing differential equations and hence, the exact elemental dynamic stiffness matrix is derived. Thus, in the absence of any discontinuities, one element is sufficient to model 1-D waveguide of any length. This elemental stiffness matrix can be assembled to obtain the global matrix as in FEM, but in the transformed space. Thus after obtaining the solution, the original domain responses are obtained using the inverse transform. Both the above mentioned manuscripts present the Fourier transform based spectral finite element (FSFE), which has the inherent aliasing problem that is persistent in the application of the Fourier series/Fourier transforms. This is alleviated by using an additional throw-off element and/or introducing slight damping in to the system. More recently wave let transform based spectral finite element(WSFE) has been formulated which alleviated the aliasing problem; but has a limitation in obtaining the frequency characteristics, like the group speeds are accurate only up-to certain fraction of the Nyquist(central frequency). Currently in this thesis Laplace transform based spectral finite elements(LSFE) are developed for sandwich members. The advantages and limitations of the use of different transforms in the spectral ﬁnite element framework is presented in detail in Chapter-1.
Sandwich structures are used in the space craft industry due to higher stiffness to weight ratio. Many issues considered in the design and analysis of sandwich structures are discussed in the well known books(by Zenkert, Beitzer). Typically the main load bearing structures are modeled as beam sand plates. Plate structures with kh<1 is analysed based on the Kirch off plate theory/Classical Plate Theory(CPT) and when the bending wavelength is small compared to the plate thickness, the effect of shear deformation and rotary inertia needs to be included where, k is the wave number and h is the thickness of the plate. Many works regarding the wave propagation in sandwich structures has been published in the past literature for wave propagation in infinite sandwich structure and giving the complete description of dispersion relation with no restriction on frequency and wavelength. More recently exact analytical solution or simply supported sandwich plate has been derived. Also it is seen by comparison of dispersion curves obtained with exact (3D formulation of theory of elasticity) and simplified theories (2D formulation as generalization of Timoshenko theory) made on infinite domain and concluded that the simplified theory can be reliably used to assess the waveguide properties of sandwich plate in the frequency range of interest. In order to approach the problems with finite domain and their implementation in the use of general purpose code; finite degrees of freedom is enforced. The concept of displacement based theories provides the flexibility in assuming different kinematic deformations to approach these problems. Many of the displacement based theories incorporate the Equivalent Single Layer(ESL) approach and these can capture the global behavior with relative ease. Chapter-2 presents the Laplace spectral finite element for thick beams based on the First order Shear Deformation Theory (FSDT). Here the effect of different choices of the real part of the Laplace variable is demonstrated. It is shown that the real part of the Laplace variable acts as a numerical damping factor. The spectrum and dispersion relations are obtained and the use of these relations are demonstrated by an example. Here, for sandwich members based on FSDT, an appropriate choice of the correction factor ,which arises due to the inconsistency between the kinematic hypothesis and the desired accuracy is presented. Finally the response obtained by the use of the element is validated with experimental results.
For high shock loading cases, the core flexibility induces local effects which are very predominant and this can lead to debonding of face sheets. The ESL theories mentioned above cannot capture these effects due to the computation of equivalent through the thickness section properties. Thus, higher order theories such as the layer-wise theories are required to capture the local behaviour. One such theory for sandwich panels is the Higher order Sandwich Plate theory (HSaPT). Here, the in-plane stress in the core has been neglected; but gives a good approximation for sandwich construction with soft cores. Including the axial inertial terms of the core will not yield constant shear stress distribution through the height of the core and hence more recently the Extended Higher order Sandwich Plate theory (EHSaPT) is proposed. The LSFE based on this theory has been formulated and is presented in Chapter-4. Detailed 3D orthotropic properties of typical sandwich construction is considered and the core compressibility effect of local behavior due to high shock loading is clearly brought out. As detailed local behavior is sought the degrees of freedom per element is high and the specific need for such theory as compared with the ESL theories is discussed.
Chapter-4 presents the spectral finite element for plates based on FSDT. Here, multi-transform method is used to solve the partial differential equations of the plate. The effect of shear deformation is brought out in the spectrum and dispersion relations plots. Response results obtained by the formulated element is compared and validated with many different experimental results.
Generally structures are built-up by connecting many different sub-structures. These connecting members, called joints play a very important role in the wave transmission/attenuation. Usually these joints are modeled as rigid joints; but in reality these are flexible and exhibits non-linear characteristics and offer high damping to the energy flow in the connected structures. Chapter-5 presents the attenuation and transmission of wave energy using the power flow approach for rigid joints for different configurations. Later, flexible spectral joint model is developed and the transmission/attenuation across the flexible joints is studied.
The thesis ends with conclusion and highlighting futures cope based on the developments reported in this thesis.2017-12-11T18:30:00ZWave Propagation in Sandwich Beam Structures with Novel Modeling Schemes
http://etd.iisc.ernet.in/2005/2737
Title: Wave Propagation in Sandwich Beam Structures with Novel Modeling Schemes
Authors: Sudhakar, V
Abstract: Sandwich constructions are the most commonly used structures in aircraft and navy industries, traditionally. These structures are made up of the face sheets and the core, where the face sheets will be taking the load and is connected to other structural members, while the soft core material, will be used to absorb energy during impact like situation. Thus, sandwich constructions are mainly employed in light weight structures where the high energy absorption capability is required. Generally the face sheets will be thin, made up of either metallic or composite material with high stiffness and strength, while the core is light in weight, made up of soft material. Cores generally play very crucial role in achieving the desired properties of sandwich structures, either through geometric arrangement or material properties or both. Foams are in extensive use nowadays as core material due to the ease in manufacturing and their low cost. They are extensively used in automotive and industrial field applications as the desired foam density can be fabricated by adjusting the mixing, curing and heat sink processes.
Modeling of sandwich beams play a crucial role in their design with suitable finite elements for face sheets and core, to ensure the compatibility between degrees of freedom at the interfaces. Unless the mathematical model simulates the physics of the model in terms of kinematics, boundary and loading conditions, results predicted will not be accurate. Accurate models helps in obtaining an efficient design of sandwich beams. In Structural Health Monitoring studies, the responses under the impact loading will be captured by carrying out the wave propagation analysis. The loads applied will be for a shorter duration (in the orders of micro seconds), where higher frequency modes will be excited. Wavelengths at such high frequencies are very small and hence, in such cases, very fine mesh generally is employed matching the wavelength requirement of the propagating wave. Traditional Finite element softwares takes enormous time and computational e ort to provide the solution. Various possible models and modeling aspects using the existing Finite element tools for wave propagation analysis are studied in the present work.
There exists a huge demand for an accurate, efficient and rapidly convergent finite elements for the analysis of sandwich beams. E orts are made in the present work to address these issues and provide a solution to the sandwich user community. Super convergent and Spectral Finite sandwich Beam Elements with metallic or composite face sheets and soft core are developed. As a philosophy, the sandwich beam finite element is constructed with the combination of two beams representing the face sheets (top and bottom) at their neutral axis. The core effects are captured at the interface boundaries in terms of shear stress and normal transverse stress.
In the case of wave propagation analysis, the equations are coupled in time domain and spatial domain and solving them directly is a difficult task. In Spectral Finite Element Method(SFEM), the displacement functions are derived by solving the transformed governing equations in the frequency domain. By transforming them and forces from time domain to frequency domain, the coupled partial differential equations will become coupled ordinary differential equations. These equations in frequency domain, can be solved exactly as they are normally ordinary differential equation with constant coefficients with frequency entering as a parameter. These solutions will be used as interpolating functions for spectral element formulation and in this respect it differs from conventional FE method wherein mostly polynomials are used as interpolating functions. In addition, SFEM solutions are expressed in terms of forward and backward moving waves for all the degrees of freedom involved in the formulations and hence, SFEM provides faster and efficient solutions for wave propagation analysis.
In the present work, strong form of the governing differential equations are derived for a given system using Hamilton's principle. Super Convergent elements are developed by solving the static part of the governing differential equations exactly and hence the stiffness matrix derived is exact for point static loads. For wave propagation analysis, as the mass is not exactly represented, these elements are required in the optimal numbers for getting good results. The number of these elements required are generally much lesser than the number of elements required using traditional finite elements since the stiffness distribution is exact. Spectral elements are developed by solving the governing equations exactly in the frequency domain and hence the dynamic stiffness matrix derived is exact for the dynamic loads. Hence, one element between any two joints is enough to solve the whole system under impact loads for simple structures.
Developing FE for sandwich beams is quiet challenging. Due to small thickness, the face sheets can be modeled using 1D idealization, while modeling of large core requires 2-D idealization. Hence, most finite or spectral elements requires stitching of these two idealizations into 1-D idealization, which can be accomplished in a variety of ways, some of which are highlighted in this thesis.
Variety of finite and spectral finite elements are developed considering Euler and Timoshenko beam theories for modeling the sandwich beams. Simple element models are built with rigid core in both the theories. Models are also developed considering the flexible core with the variation of transverse displacements across depth of the core. This has direct influence on shear stress variation and also transverse normal stress in the core. Simple to higher order models are developed considering different variations in shear stress and transverse normal stress across depth of the core. Development of super convergent finite Euler Bernoulli beam elements Eul4d (4 dof element), Eul10d (10 dof element) are explained along with their results in Chapter 2. Development of different super convergent finite Timoshenko beam elements namely Tim4d (4 dof), Tim7d (7 dof), Tim10d (10 dof) are explained in Chapter 3. Validation of Euler Bernoulli and Timoshenko elements developed in the present work is carried out with test cases available in the open literature for displacements and free vibration frequencies are presented in Chapter 2 and Chapter 3. The results indicates that all developed elements are performing exceedingly well for static loads and free vibration. Super convergence performance for the elements developed is demonstrated with related examples.
Spectral elements based on Timoshenko theory STim7d, STim6d, STim6dF are developed and the wave propagation characteristics studies are presented in Chapter 4. Euler spectral elements are derived from Timoshenko spectral elements by enforcing in finite shear rigidity, designated as SEul7d, SEul6d, SEul6dF and are presented. E orts were made in this present work to model the horizontal cracks in top or bottom face sheets using the spectral elements and the methodology is presented in Chapter 4.
Wave propagation analysis using general purpose software N AST RAN and the super convergent as well as spectral elements developed in this work, are discussed in detail in Chapter 5. Modeling aspects of sandwich beam in N AST RAN using various combination of elements available and the performance of four possible models simulated were studied. Validation of all four models in N AST RAN, Super convergent Euler, Timoshenko and Spectral Timoshenko finite elements was carried out by simulating a homogenous I beam by comparing the longitudinal and transverse responses. Studies were carried out to find out the response predictions of a sandwich beam with soft core and all the predictions were compared and discussed. The responses in case of cracks in top or bottom face sheets under the longitudinal and transverse loading were studied in this chapter.
In Chapter 6, Parametric studies were carried out for bringing out the sensitiveness of the important specific parameters in overall behaviour and performance of a sandwich beam, using Super convergent and Spectral elements developed. This chapter clearly brings out the various aspects of design of sandwich beam such as material selection of core, geometrical configuration of overall beam and core. Effects of shear modulus, mass density on wave propagation characteristics, effects of thick or thin cores with reference to the face sheets and dynamic effects of core are highlighted. Wave propagation characteristics studies includes the study of wave numbers, group speeds, cut off frequencies for a given configuration and identification of frequency zone of operations. The recommendations for improvement in design of sandwich beams based on the parametric studies are made at the end of chapter.
The entire thesis, written in seven Chapters, presents a unified treatment of sandwich beam analysis that will be very useful for designers working in the area.2017-10-30T18:30:00Z