etd@IISc Collection:
http://hdl.handle.net/2005/27
Wed, 23 Apr 2014 08:15:39 GMT2014-04-23T08:15:39ZThe Channel Imagehttp://etd.ncsi.iisc.ernet.in:80/retrieve/27/aerospace.jpg
http://hdl.handle.net/2005/27
Weighted Least Squares Kinetic Upwind Method Using Eigendirections (WLSKUM-ED)
http://hdl.handle.net/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.Wed, 24 Jun 2009 07:07:29 GMThttp://hdl.handle.net/2005/5382009-06-24T07:07:29ZWave Propagation In Anisotropic & Inhomogeneous Structures
http://hdl.handle.net/2005/1138
Title: Wave Propagation In Anisotropic & Inhomogeneous Structures
Authors: Chakraborty, AbirThu, 28 Apr 2011 18:30:00 GMThttp://hdl.handle.net/2005/11382011-04-28T18:30:00ZWavelet Based Spectral Finite Elements For Wave Propagation Analysis In Isotropic, Composite And Nano-Composite Structures
http://hdl.handle.net/2005/448
Title: Wavelet Based Spectral Finite Elements For Wave Propagation Analysis In Isotropic, Composite And Nano-Composite Structures
Authors: Mitra, Mira
Abstract: Wave propagation is a common phenomenon in aircraft structures resulting from high velocity transient loadings like bird hit, gust etc. Apart from understanding the behavior of structures under such loading, wave propagation analysis is also important to gain knowledge about their high frequency characteristics, which have several applications. The applications include structural health monitoring using diagnostic waves and control of wave transmission for reduction of noise and vibration.
Transient loadings with high frequency content are associated with wave propagation. As a result, the higher modes of the structure participate in the response. Finite element (FE) modeling for such problem requires very fine mesh to capture these higher modes. This leads to large system size and hence large computational cost. Wave propagation problems are usually solved in frequency domain using fast Fourier transform (FFT) and spectral finite element method is one such technique which follows FE procedure in the transformed frequency domain.
In this thesis, a novel wavelet based spectral finite element (WSFE) is developed for wave propagation analysis in finite dimension structures. In WSFE for 1-D waveguides, the partial differential wave equations are reduced to a set of ODEs using orthogonal compactly supported Daubechies scaling functions for temporal approximation. The localized nature of the Daubechies basis functions allows finite domain analysis and imposition of the boundary conditions. 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 equation 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 FE and after solution, the time domain responses are obtained using the inverse wavelet transform.
The developed technique circumvents several serious limitations of the conventional FFT based Spectral Finite Element (FSFE). In FSFE, the wave equations are reduced to ODEs using FFT for time approximation. The remaining part of the formulation is quite similar to that of WSFE. The required assumption of periodicity in FSFE, however, does not allow modeling of finite length structures. It results in “wrap around” problem, which distorts the response simulated using FSFE and a semi-infinite (“throw-off”) element is required for imparting artificial damping. This artificial damping occurs as the “throw off” element allows leakage of energy. In some cases, a very high damping can also be considered instead of “throw off” element to remove wrap around effects. In either cases, the damping introduced is much larger than any inherent damping that may be present in the structure. It should also be mentioned that even in presence of the artificial damping, a larger time window is required for removing the distortions completely. The developed WSFE method is completely free from such problems and can efficiently handle undamped finite length structures irrespective of the time window considered. Apart from this, FSFE allows imposition of only zero initial condition and in contrary any initial conditions can be used in WSFE.
Though FSFE has problem in modeling finite length undamped structures for time domain analysis, it is well suited for performing frequency domain study of wave characteristics, namely, the determination of spectrum and dispersion relations. WSFE is also capable of extracting these frequency dependent wave properties, however only up to a certain fraction of the Nyquist frequency. This constraint results from the loss in frequency resolution due to the increase in time resolution in wavelet analysis, where the basis functions are bounded both in time and frequency. A price has to be paid in frequency domain in order to obtain a bound in the time domain. The consequence of this analysis is to impose a constraint on the time sampling rate for the simulation with WSFE, to avoid spurious dispersion.
WSFE for 2-D waveguides are formulated using Daubechies scaling functions for both temporal and spatial approximations. The initial and boundary conditions, however, are imposed using two different methods, which are wavelet extrapolation technique and periodic extension or restraint matrix respectively. The 2-D WSFE is bounded in both the spatial directions unlike 2-D FSFE, which is essentially unbounded in one spatial direction. Apart from this, 2-D WSFE is also free from “wrap around” problem similar to 1-D WSFE due to the localized nature of the basis functions used for temporal approximation.
In this thesis, WSFE is developed for isotropic 1-D and 2-D waveguides for time and frequency domain analysis. These include elementary rod, Euler-Bernoulli and Timoshenko beams in 1-D modeling, and plates and axisymmetric cylinders in 2-D modeling. The wave propagation responses simulated using WSFE for these waveguides are validated using FE results. The advantages of the proposed technique over the corresponding FSFE method are also highlighted all through the numerical examples.
Next part of the thesis involves the extension of the developed WSFE technique for modeling composite and nano-composite structures to study their wave propagation behavior. Due to their anisotropic nature, analysis of composite structures, particularly high frequency transient analysis is much more complicated compared to the corresponding metallic structures. This is due to the presence of stiffness coupling in these structures. Superior mechanical properties of composites, however, are making them integral parts of an aircraft and thus they often experience such short duration, high velocity impact
Loadings. Very few literatures report the response of composite structures subjected to such high frequency excitations. Here, WSFE is formulated for a higher order composite beam with axial, flexural, shear and contractional degrees of freedom. WSFE is also formulated for composite plates using classical laminated plate theory with axial and flexural degrees of freedom. Simulations performed using these WSFE models are used to study the higher order and elastic coupling effects on the wave propagation responses.
Carbon nanotubes (CNTs) and their composites are attracting a great deal of experimental and theoretical research world-wide. The recent trend in the literature shows a great interest in the dynamic and wave characteristics of CNTs and nano-composites because of their several applications. In most of these applications, CNTs are used in the embedded form as it does not requires precise alignment of the nano-tubes. In addition, the extraordinary mechanical properties of CNTs are being exploited to achieve high strength nano-composite. Apart from the experimental studies and atomistic simulation to study the mechanical properties of CNTs and nano-composites, continuum modeling is also receiving much attention, mainly due to its computational viability. In this thesis, a 1-D WSFE is formulated for multi-wall carbon nanotube (MWNT) embedded composite modeled as beam using higher order layer-wise theory. This theory allows to model partial interfacial shear stress transfer, which normally occurs due to improper dispersion of CNTs in nano-composites. The effects of different matrix materials and fraction of shear stress transfer on the wave characteristics are studied. The responses obtained using other beam theories are also compared.
The beam modeling does not allow capturing the radial motions of the CNT, which are important for several applications. These can be effectively captured by modeling the CNT using a 2-D axisymmetric model. Hence, a 2-D WSFE model is constructed to capture the high frequency characteristics of single-walled carbon nanotubes (SWNTs). The response of SWNT simulated using the developed model is validated with experimental and atomistic simulation results reported in the literature. The comparison are done for dispersion relation and also radial breathing mode frequencies. The effects of geometrical parameters, namely the radius and the wall thickness of the SWNT on the higher radial, longitudinal and coupled radial-longitudinal vibrational modes are analyzed. These behaviors are studied in both time and frequency domains. Such time domain analyses of finite length SWNT are not possible with the Fourier transform based techniques reported in literature, although, such analyses are important particularly for sensor applications of SWNT.
Spectral finite element method is very much suited for solution of inverse problems like force reconstruction from the measured wave response. This is because the technique is based on the concept of transfer function between the displacements (output) and applied forces (input). In the present work, WSFE is implemented for identification of impact force from the wave propagation responses simulated with FE and used as surrogate experimental results. The results show that WSFE can accurately reconstruct the impulse load applied to 1-D waveguides which include rod, Euler-Bernoulli beam and connected 2-D frame, even with highly truncated response. This is unlike FSFE, where the accuracy of the identified force depends largely on the time window of the measured responses.
The detection of damage from the wave propagation analysis is another class of inverse problems considered in this thesis and is of utmost importance in the area of aircraft structural health monitoring. Here, the detection scheme is based on arrival time of the waves reflected from the damage. A novel detection technique based on wavelet filtering is proposed here and it is shown to work efficiently even in the presence of noise in the measured wave responses. Detection of damage requires an efficient damage model to simulate the mode of structural failure. In this regard, two spectrally formulated wavelet elements are proposed, one to model isotropic beam with through-width notch and the second to model composite beam with embedded de-lamination. In the first case, the response of the damaged beam is considered as the perturbation of the undamaged response and the linear perturbation analysis leads to a completely new set of dynamic stiffness matrix. In the second case, the delamination is modeled by subdividing the de-laminated region into separate waveguides and full damage model is established by imposing the kinematics. These models help to simulate wave propagation in such damaged beams to study the effect of damage on the wave response.
Noise and vibration are often transmitted from the source to the other parts of the structure in the form of wave propagation. Thus, control of such wave transmission is essential for reduction of noise and vibration, which are the main cause of discomfort and in many cases cause failure of structure. Here, techniques for both passive and active controls of wave are proposed. For active control, a closed loop system is modeled using WSFE with magnetostrictive actuator for control of axial and flexural wave propagations in connected isotropic 1-D waveguides. The feedback is negative velocity and/or acceleration measured at different sensor points. A very new application of CNT reinforced composite for passive control of vibration and wave response is explored in this thesis. For this, a novel concept of nano-composite inserts is proposed. This insert can be made from CNTs dispersed in polymer. The high stiffness of the inserts helps to regulate the power flow in the form of wave propagation from the point of application of the loads to other parts of the structures. The length of the insert, volume fraction of CNTs and position are changed to achieve the required reduction in wave amplitudes.
The entire thesis is split up into eight chapters. Chapter 1 presents a brief introduction, the motivation and objective of the thesis. Chapters 2 and 3 give a detail account of wavelet spectral finite element formulation for 1-D and 2-D isotropic waveguides, while Chapter 4 gives the same for composite waveguides. Chapter 5 brings out essential wave characteristics in carbon nanotubes and nano-composite structures, while Chapters 6 and 7 exclusively deal with application of WSFE to some real world problems. The thesis ends with summary and directions of future research.
In summary, the thesis has brought out several new aspects of wave propagation in isotropic, composite and nano-composite structures. In addition to establishing wavelet spectral finite element as a useful tool for wave propagation analysis, several new techniques are presented, several new algorithm are proposed and several new concepts are explored.Thu, 02 Apr 2009 09:47:53 GMThttp://hdl.handle.net/2005/4482009-04-02T09:47:53ZVibration And Impact Induced Sound
http://hdl.handle.net/2005/928
Title: Vibration And Impact Induced Sound
Authors: Narla, Subrahmanya Prasad
Abstract: Sound generated by impacting structures is of considerable importance in noise control. Sound is generated by a vibrating structure by inducing pressure fluctuations in the surrounding medium. Impact induced noise is the sound generated by a vibrating structure subjected to motion constraint. In such problems one has to study the vibration behavior of the oscillator, the impact mechanics, and the emanating acoustic field dynamics.
A literature review carried out points to the fact that though there has been considerable work on vibration behavior of impact oscillators and the acoustics of impact of rigid masses, there is very little work reported on the sound generated due to vibration and impact. This thesis couples vibration analysis of oscillators undergoing impact with its acoustic behavior. The vibration behavior is nonlinear on account of the impact. Therefore the vibration analysis as well as the resulting acoustic field analysis has to be in the time-domain. This investigation is concerned with the effect of structural dynamics, impact dynamics, and acoustic field boundary conditions, on the sound pressure generated due to vibration and impact.
We have considered a single degree of freedom as well as a flexible Euler-Bernoulli beam vibration model. The former is the simplest for studying vibro-acoustic response. The numerical model of the beam is derived using the finite element method resulting in a finite dimensional system with more than one degree of freedom. The dynamics of each degree of freedom are distinct in terms of amplitude and phase and are a function of the nature of linear dependence on other degrees of freedom and the nature of excitation. An impacting beam introduces interesting interactions between the dynamics of the degrees of freedom as a consequence of nonlinearity due to the motion constraint.
The impact of the oscillator mass with a barrier is modeled using a simple coefficient of restitution model based on Hertzian contact theory. There is velocity reversal on contact with the barrier. The contact force is finite acting within a finite interval of time. The contact force is assumed to vary in time during the contact interval. This effectively models contact as linearly elastic.
The pressure perturbation due to vibration of the oscillator mass is shown equivalent to the pressure perturbation due to an acoustic dipole. The acoustic dipole is placed at the equilibrium position of the vibrating mass. The dipole pressure is then a function of motion of the oscillator. In the case of a single degree of freedom oscillator the dipole axis is along the direction of motion. The sound pressure due to a vibrating beam is modeled as an array of acoustic dipoles placed at the finite element nodes of the beam and stationary at the beam's static equilibrium configuration. The dipole axis is once again aligned with the direction of vibration of the beam that is transverse to the beam neutral axis. Anechoic as well as perfectly reflecting acoustic boundary conditions are simulated in the time-domain.
The resulting governing equation of motion of the single degree freedom oscillator as well as the beam are integrated numerically in time to compute its response.
The acoustic pressure is shown to be critically dependent on the excitation frequency of the oscillator, dynamic properties of the oscillator, coefficient of restitution of impact and impact dynamics, and acoustic field boundary conditions.Thu, 21 Oct 2010 18:30:00 GMThttp://hdl.handle.net/2005/9282010-10-21T18:30:00Z