Mandar D. Kulkarni

Stress wave attenuation in ceramic plates

Studies on planar longitudinal stress wave attenuation in thin ceramic plates are presented using a wave tracking algorithm. The results are also experimentally validated. Ceramic plates consist of grains and grain boundaries. Elastic modulus and density of the grain boundaries are less than those of the grains. When the planar longitudinal stress wave encounters an interface between the grain and the grain boundary, reflection and transmission of the incident stress wave take place because of the impedance mismatch at the interface. It is observed that this leads to attenuation in stress wave intensity. This behavior is represented by stress wave attenuation coefficient. It is observed that the stress wave attenuation coefficient is a material property.

Discrete Adjoint Formulation for Continuum Sensitivity Analysis

Continuum Sensitivity Analysis (CSA) is an approach for calculating analytic derivatives. A direct CSA formulation is advantageous for computing derivatives of many state variables or performance functions. An adjoint formulation of CSA is beneficial for computing derivatives with respect to many design variables, although adjoint CSA boundary conditions are often problematic. For the proposed continuum-discrete hybrid adjoint approach, the adjoint variable is introduced after discretization which simplifies boundary conditions. The sensitivity boundary conditions for the hybrid CSA are posed in terms of the continuum state variables. Thus, the hybrid adjoint formulation of CSA results in design derivatives that are as accurate as those obtained from direct CSA, in addition to making the analysis efficient for the case of large number of design variables. Two test cases, first of an axial bar and second of a cantilever beam modeled with solid elements, illustrate how the hybrid adjoint formulation inherits the benefits of the direct and adjoint CSA formulations. This is also the first application of CSA to obtain design derivatives nonintrusively using three dimensional (3-D) Spatial Gradient Reconstruction (SGR) method for 3-D solid elements.

Integration of Geometric Sensitivity and Spatial Gradient Reconstruction for Aeroelastic Shape Optimization

Continuum Sensitivity Analysis (CSA) provides an analytic method to obtain sensitivities of structural and fluid responses. Its primary advantages are that sensitivities are analytic and mesh sensitivity is avoided. CSA involves solving a set of Continuum Sensitivity Equations (CSEs) derived from the governing equations of the original analysis. Solution to CSE requires two terms: spatial gradients of the response and geometric sensitivity, also known as design velocity. Spatial gradients of the response may be obtained using Spatial Gradient Reconstruction (SGR) technique. In this paper we present complex step results for computing the geometric sensitivity from tools such as MstcGeom and VT-CST. Use of SGR in CSA makes amenable the non-intrusive implementation of CSA. In this paper, we describe the specific requirements of a fluid analysis code for non-intrusive implementation of CSA. The results of a survey of some flow solvers are presented that help choose a flow solver. Flow sensitivities of a lid-driven cavity are computed using CSA. This example illustrates that flow sensitivities of a nonlinear fluid system can be found accurately by CSA with just a change of the boundary conditions and source terms, while retaining the same discretization. Another example is presented to illustrate the use of the flow solver Stanford University Unstructured (SU2) for obtaining aeroelastic shape sensitivity. This example establishes the use of SU2 for aeroelastic analysis and to obtain its sensitivity by the CSA approach.

Nonintrusive Continuum Sensitivity Analysis for Aerodynamic Shape Optimization

Continuum Sensitivity Analysis (CSA) provides an analytic method to obtain derivatives of structural and fluid responses. The primary advantages of the presented local formulation are that analytic derivatives are computed and mesh sensitivity is avoided. With the use of Spatial Gradient Reconstruction technique, it has been shown that nonintrusive implementation of CSA is possible for structural systems. In the current work, we extend this to fluid systems. The specific requirements of a computational fluid dynamics solver for such a nonintrusive implementation are described in detail. Derivatives of flow variables have been calculated using CSA for two examples, (a) flow in a lid-driven cavity, and (b) subsonic-supersonic isentropic flow through a convergent divergent nozzle. The corresponding sensitivity equations and boundary conditions are derived.

Reliability Based Structural Design using Continuum Sensitivity Analysis

Reliability based design of structural systems is necessary to account for the randomness in loads, structural geometry, material properties, manufacturing processes, and operational environment. Probabilistic methods are commonly used for reliability based design. Gradients of the objective and limit state functions that are required during stochastic optimization procedure are almost always calculated using finite dierence method. However, calculating gradients in this way is very computationally expensive. In this paper, we propose a stochastic optimization procedure, which makes use of continuum sensitivity analysis for obtaining gradients. It is expected that this would not only result in significant savings in computational eorts, but also accurate design derivatives than those obtained with other numeric design sensitivity analysis methods. Another part of the current work is to investigate the most appropriate polynomial family and order of expansion to use for polynomial chaos expansion. The use of Jacobi and Hermite polynomials is compared. It is seen that if the input variables follow a beta distribution, then second order Jacobi polynomials give the best stochastic optimization result.

Reduced Order Model for Unsteady Aerodynamics of Flapping Wing Micro Air Vehicle in Hover

Aerodynamics of a flapping wing Micro Air Vehicle (MAV) in hover is highly unsteady. Wake shed by the airfoil remains close to the airfoil surface. In this case, using low fidelity quasi-steady aerodynamic models does not give good estimate of lift and drag forces. Also, Momentum Disc Theory (MDT) alone cannot be used to model the inflow for such a complicated wake. High fidelity methods such as Computational Fluid Dynamics (CFD) are too computationally expensive for doing optimization and sensitivity studies for flapping MAVs. So, medium fidelity tools such as Unsteady Vortex Lattice Method (UVLM) have been used for modeling inflow of flapping wing mainly for forward flight conditions. However, presently there are no reduced order threedimensional (3D) aerodynamic models which can be used for doing preliminary design studies for flapping wing MAVs in hover. In the present work, a reduced order scheme is proposed which uses MDT and UVLM for modeling inflow of a flapping 2D airfoil. Results indicate that retaining only a fraction of the shed vortices is sucient to get reasonable accuracy. For example, retaining vortices shed in the recent two out of ten oscillations reduced the error in lift by 88% as compared to quasi-steady calculation, i.e. it captures 88% of the unsteadiness. Addition of inflow calculated using MDT along with retaining two oscillations, helps capture 92% of the unsteadiness. Furthermore, the proposed scheme also helps capture about 90% of the unsteadiness in drag calculations. Since the proposed scheme simplifies computation significantly, it can be extended to create a 3D aerodynamic model for flapping wing MAV in hover.