The numerical solution of time-dependent systems is often carried out using a step-wise updating procedure known as marching in time or simply time-marching methods. They require completing previous steps to yield an update for the next time step. This imposes a barrier in implementing the algorithm in a computer program, which prevents parallel speed-up by increasing the serial fraction of the program. This paper analyzes a method based on the space-time discretization of a time-dependent system using Discrete Picard Iterations (DPI). Unlike time marching schemes, the updating barrier is removed in the current formulation by converting the solution procedure into a set of larger matrix–vector products that correspond to the simultaneous computation of spatial residuals at various time levels. The stability and accuracy of the proposed method are investigated. The accuracy and computation time are compared to various popular time-marching methods involving structured and higher order/spectral unstructured grids in one and two-dimensional space. It is shown that when the inverse of the Chebyshev differentiation matrix is used in the DPI framework, the resulting scheme is superior in both accuracy and CPU time. Various forms of iterative methods and preconditionings are implemented and evaluated. This includes GMRES, IDR(s), and Conjugate Gradient methods. It is shown that the stabilized Bi-conjugate gradient method produces the best iterative solver when a band-LU preconditioner is used. This method, which relies heavily on matrix multiplication, is strategically consistent with the recent development in the hardware implementation of matrix multiplication accelerators on ASICs and FPGAs.

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed.

In the design and manufacturing of sensitive crane mechanisms, in particular, those utilized in nuclear facilities, safety is of utmost importance. A few designs are based on a hydraulic equalizing cylinder where the purpose is mainly to dampen the dynamic load factor generated by the sudden failure of the wire rope system. To the best of the authors' knowledge, design equations and engineering models for such an intricate dynamic mechanism are not reported anywhere in the literature. Toward achieving this goal, a model is proposed for the dynamic behavior of a hydraulic equalizing cylinder when coupled to a single-failure-proof cross-reeved crane hoist. The constitutive equations involve the time-dependent incompressible Navier-Stokes equations. These equations are coupled with the dynamics of the lower block and the elastic wire rope. A nonlinear second-order ordinary differential equation is obtained, which is solved analytically. This yields a closed-form relation for the dynamic load factor. The validity of the new model is investigated using experimental studies, which are presented in this work. The normalized root mean square error for the dynamic load factor obtained by the model compares favorably to the experimental results, where the maximum observed error is 3.9 percent. The results confirm that the current model can be used as a design equation by practitioners.

In this paper, the accuracy of two mesh-free approximation approaches, the Gravity model and Radial Basis Function, are compared. The two schemes' convergence behaviors prove that RBF is faster and more accurate than the Gravity model. As a case study, the interpolation of temperature at different locations in Tennesse, USA, are compared. Delaunay mesh generation is used to create random points inside and on the border, which data can be incorporated in these locations. 49 MERRA weather stations as used as data sources to provide the temperature at a specific day and hour. The contours of interpolated temperatures provided in the result section assert RBF is a more accurate method than the Gravity model by showing a smoother and broader range of interpolated data.

In this paper, the accuracy of two mesh-free approximation approaches, the Gravity model and Radial Basis Function, are compared. The two schemes' convergence behaviors prove that RBF is faster and more accurate than the Gravity model. As a case study, the interpolation of temperature at different locations in Tennesse, USA, are compared. Delaunay mesh generation is used to create random points inside and on the border, which data can be incorporated in these locations. 49 MERRA weather stations as used as data sources to provide the temperature at a specific day and hour. The contours of interpolated temperatures provided in the result section assert RBF is a more accurate method than the Gravity model by showing a smoother and broader range of interpolated data.

Grid generation is an important and challenging preprocessing step for setting up a detailed discrete hydrodynamic model for simulating river flows. Often, resolution of the modeling study is determined by limitations of the grid generation programs. Also, compatibility of grid generators to match with a hydrodynamic model is another challenge. Environmental Protection Agency's Environmental Fluid Dynamics Code (EFDC) is one of the most popular models used for simulating surface water hydrodynamics and water quality. However, it's associated Fortran-based GEFDC grid generator for EFDC is complex and requires a substantial amount of mathematical knowledge and craftsmanship to set up a compatible grid. Taking this into consideration, we developed a new algorithm and implemented in a MATLAB based grid generator program that automates reproducible grid generation from a variety of data source formats. As part of this paper, we present the grid generation process using a test model that was set up and executed to produce results and verify compatibility of our grid generator with EFDC model.

................................................................................................................................. iii DEDICATION ............................................................................................................................... iv ACKNOWLEDGEMENTS ............................................................................................................ v LIST OF TABLES ....................................................................................................................... viii LIST OF FIGURES ....................................................................................................................... ix LIST OF ABBREVIATIONS ....................................................................................................... xi LIST OF SYMBOLS .................................................................................................................... xii

1Graduate Transportation Associate, Tennessee Department of Transportation 2Associate Professor, University of Tennessee at Chattanooga 3Transportation Engineer, Maryland Department of Transportation 4Civil Engineering Manager, Tennessee Department of Transportation 5Director, Civil Infrastructure Laboratory & Research, University of Tennessee at Chattanooga ---------------------------------------------------------------------***---------------------------------------------------------------------Abstract Structural strength assessment of civil infrastructure is a prior concern to engineers. This paper presents laboratories study as well as on site study data analysis gathered by many authors and critical review of existing nondestructive testing (NDT) methods. The techniques considered are ultrasonic pulse velocity method, surface hardness method, penetration resistant method, pullout test, the impact-echo system, infrared thermography, ground penetration radar, ultrasonic methods, acoustic emission method, maturity method, stress wave propagation method, corrosion potential test, X-ray computed tomography, Electrical Impedance Spectroscopy (EIS), and Wireless sensor network. Applications, advantages, accuracy, limitations, instrumentation, principles, and ASTM standard of these techniques are discussed for the infrastructural condition assessment.

The problem of preconditioning the linear system Ax = b where no entries of A are known but the matrix-vector product is given in the linear functional form Ax = L(x) is considered. A statistical multiregressor method is proposed whose convergence improves during Krylov iterations. It is shown that this method effectively improves the rate of convergence of the GMRES algorithm. The results are validated by applying the current preconditioner to a pseudo-spectral solution of Helmholtz equation.

In conventional spectral/finite element methods, the triangulation/quadrilateralization of the domain produces many interior edges which require additional DOF. What if we could directly use the original hull without going to triangulation/quadrilateralization? There are three major difficulties with this type of approach that are addressed: 1- How can a convex hull tessellation be obtained for complex geometries encountered in practical engineering applications? 2- How can basis functions and quadrature points be defined on these hulls? 3- How can this type of grid and corresponding basis functions be used in practice to yield accurate discretization of nonlinear conservation laws? Geometrical approaches to tackle the first challenge are discussed . To solve the second challenge, for the first time in the literature, a closed form relation is proposed to approximate Fekete points (a Nondeterministic Polynomial (NP) problem) on a general convex/concave polyhedral. The approximate points are used to generate basis functions using the SVD of the Vandermonde matrix. The type of basis functions derived include Lagrange basis, orthogonal and orthonormal hull basis and radial basis. It is shown that the hull basis is the best choice to enforce minimum DOF while maintaining a small Lebesgue constant when very high p-refinement is done. The proposed hull basis is rigorously proven to achieve arbitrary order of accuracy by satisfying Weierstrass approximation theorem in $\mathbb{R}^d$. The third challenge is met using a standard Discontinuous Galerkin (DG) and Discontinuous Least-Squares (DLS) spectral hull formulations. The accuracy and stability of the formulation is demonstrated for the linearized acoustics and two-dimensional compressible Euler equations on some benchmark problems including a cylinder, airfoil, vortex convection and compressible vortex shedding from a triangle.

A scalable paradigm is developed to generate 2D/3D high quality finite/spectral element meshes containing arbitrarily curved elements. The current methodology begins with a linear mesh that is decomposed using a graph partitioning scheme. Higher-order elements are then created from the linear mesh, where a CAD model must be queried in order for the curved faces/edges to conform to the boundaries. Subsequently, the curved elements are directly generated using analytical maps which transform the point distribution of the master element to the element in physical space. These analytic maps are derived for triangular, quadrilateral, tetrahedral, prismatic, pyramidal, and hexahedral elements. It is shown that the stretching of Chebyshev/Fekete point distributions are also preserved by these maps and hence they can be used to generate well-conditioned spectral element grids. Since these maps require a computationally intensive min-distance projection to the CAD model, a fast min-distance search algorithm is proposed. The current method is embarrassingly parallel, uses MPI, and is implemented on a commodity cluster. Degradation in performance is observed with load balancing based on maximizing the volume to surface ratio and, therefore, a new load balancing is proposed to mitigate this loss in speed-up. Results are presented for a two-dimensional cylinder, NACA0012 airfoil, 30P30N high-lift geometry, a three-dimensional sphere, a notional missile configuration and a civil aircraft geometry.

To predict allowable time-step size for the fully discretized nonlinear differential equations, a stability theory is developed using exact determination of an infinite perturbation series. Mathematical induction is used to determine the coefficients of the series. It is discovered that the closed-form equation for the nonlinear shift of generic polynomial non-linearity can be written as a series expansion where the coefficients are the Pfaff-Fuss-Catalan numbers in Combinatorics. This reveals criteria which can be used to analytically determine the allowable time step. It is shown that stability region decreases when the nonlinearity of the differential equation increases. Therefore, the maximum allowable time step is severely limited by the nonlinearity even if an unconditionally stable scheme (in a linear sense) is used. The theory is applied to general system of time-dependent nonlinear Partial Differential Equations.

Preconditioning of a linear system obtained from spectral discretization of time-dependent PDEs often results in a full matrix which is expensive to compute and store specially when the problem size increases. A matrix-free implementation is usually applied to resolve this issue. In this framework, preconditioning is typically challenging since the entries of the matrix are not explicitly available. In this short note, we propose a statistical approach to gradually create a preconditioner matrix by collecting the information obtained from matrix-vector product in the Arnoldi loop of an unpreconditioned Krylov subspace algorithm. The gathered information are then correlated using a multiple regressors estimate where the error is assumed to be normally distributed. This procedure yields a banded diagonal matrix which is then used as a preconditioner in the next iterative solve. This is repeated between iterative solves until a good preconditioner is constructed on fly. This statistically iterative procedure is progressive since the fidelity of the preconditioning matrix improves by adding more data obtained from matrix-vector product during the entire solution procedure. The proposed algorithm is validated for a sample implementation.

An iterative numerical time marching algorithm which is applicable for solving time dependent anisotropic nonlinear Maxwell equations is presented. The method is based on high-order discretization of classical Picard iteration. Using linearization in the iteration space, a highly convergent implicit formulation is derived. The dissipation error is completely eliminated when Picard iteration converges since no linearization is done in time. In addition, the order of accuracy in space and time can be increased arbitrarily without violating unconditional stability of the scheme. Numerical test cases are performed to validate the convergence and accuracy of the proposed method.

This paper presents the derivation and implementation of a new family of high-resolution compact schemes and implicit Picard-like iterative algorithms for solving sti nonlinear system of ODEs. We rst derive compact nite dierence schemes for integration which can be regarded as the missing part of Lele’s work(J. Comp. Physics, 103, pp. 16{42.). In addition to the interior scheme, proper discretization for points near boundaries is obtained. Near boundary schemes are tabulated based on the order of accuracy and the location where scheme is used. Spectral analysis is performed and a general relation for the modied wave number is obtained. In addition to theoretical analysis of the order of accuracy, numerical studies are performed and the results are compared to other integrators including trapezoidal, Riemann series, and the classical fourth-order Runge-Kutta scheme. Grid renement study reveals competitive convergence of compact schemes for integration. In the second part of the paper, we proceed to reformulate the classic Picard iteration theory in the discretized form using the developed compact schemes. An interesting convergence condition is obtained. A rigorous mathematical analysis is performed to prove that the convergence of the algorithm is exponential. Suitable Ad-hoc iteration criteria is added to accelerate the convergence while preserving the eciency. This heuristic intervalbased iterative approach is implemented for a number of famous benchmarks to observe the overall performance of the solution strategy. In the concluding section, the developed methodology is applied to a practical SixDof simulation of a tail-controlled path-tracking missile with semi-active homing guidance and PI controllers for adjusting the line of sight. The missile dynamics is fully treated using twelve equations of motions in the term of Euler angles. Aerodynamic coecients (static, dynamic, contributions by tail

Large-stencil schemes which their spectral properties are acceptable in the vicinity of ! = are analyzed for the first time. A machine independent model for evaluating the eciency of generalized time-marching finite-dierence algorithms over periodic domains is developed. This model which is based on operation count reveals that for small values of Total Computational Cost(TCC), the previous low-order small-stencil schemes are more ecient while for moderate TCC, the eciency of optimized large-stencil schemes abruptly increases. This important result is the motivation for developing optimized large-stencil schemes. The current schemes are successfully implemented in a full compressible NavierStokes solver over general coordinates. It is shown that the current schemes are stable in practical situations and are able to resolve fine structures of flow over coarse grids. The robustness of current schemes is validated through several benchmark problems of Fluid Dynamics.

Recently, several high-order accurate methods have been developed for solving nonlinear equations governing in the high-frequency oscillations of a fluid column. If we assume that the value of Reynolds and Prandtl numbers are high enough then it is possible to neglect the effect of Stokes Layer and it has been previously shown that the calculated pressure profile obtained by solving three-dimensional axisymmetric compressible NS (Navier-Stokes) equations is in good agreement with one-dimensional adiabatic model of Ilinskii. For this particular flow which is the typical of engineering flows inside acoustical compressors, it is clear that high-resolution simulation of NS equations in higher dimensions is quite expensive. For this reason, a quick method based on the lax-wendroff scheme is used in this paper for solving the one-dimensional adiabatic model. The numerical method used here is simpler in implementation than previous lagrangian and spectral approaches and is considerably more efficient than three-dimensional solutions. The results agree well with previous multi-dimensional solutions.

A new perspective suitable for understanding the details of nonlinear pumping (formation of traveling shocks) inside a pressurized cavity is constructed in this paper. Full compressible axisymmetric three-dimensional Navier-Stokes equations are used as the starting point to cover all complexities of the problem that exceedingly increase for particular ranges of Mach, Reynolds and Prandtl numbers. Then a very high-order numerical method is introduced to preserve the user-defined order of accuracy for practical simulations. For removal of spurious waves, higher-order compact filters are derived. All equations are marched in time using the classical Runge-Kutta algorithm which is appropriate for problems involving fine-scale temporal fluctuations. As the most important part of simulation, Navier-Stokes Characteristic Boundary Conditions are used for accurate calculation of wave reflection specially at singular points, i.e., corner points and points across the axis of symmetry. A simultaneous characteristic-decomposition is devised in this paper which completely resolves stability problems arising from problem-dependent treatment of corner points. Numerical experiments are performed for high-Reynolds laminar flows inside the shock region to determine the effect of frequency change on both shock formation (stationary flow) and transient solution. The current approach which favorably compares to the previous experimental data, may be used as a robust tool for understanding the less-understood problem of shock/Stokes-Layer interaction and its consequences on transition to turbulence in Oscillating Pipe Flow.