ALBERT

Note: Contents 1. Introduction 1.1 Historical Background 1.2 Some Examples of Spectral Methods 1.2.1 A Fourier Galerkin Method for the Wave Equation 1.2.2 A Chebyshev Collocation Method for the Heat Equation 1.2.3 A Legendre Galerkin with Numerical Integration (G-NI) Method for the Advection-Diffusion-Reaction Equation 1.2.4 A Legendre Tau Method for the Poisson Equation 1.2.5 Basic Aspects of Galerkin, Collocation, G-NI and Tau Methods 1.3 Three-Dimensional Applications in Fluids: A Look Ahead 2. Polynomial Approximation 2.1 The Fourier System 2.1.1 The Continuous Fourier Expansion 2.1.2 The Discrete Fourier Expansion 2.1.3 Differentiation 2.1.4 The Gibbs Phenomenon 2.2 Orthogonal Polynomials in (−1, 1) 2.2.1 Sturm-Liouville Problems 2.2.2 Orthogonal Systems of Polynomials 2.2.3 Gauss-Type Quadratures and Discrete Polynomial Transforms 2.3 Legendre Polynomials 2.3.1 Basic Formulas 2.3.2 Differentiation 2.3.3 Orthogonality, Diagonalization and Localization 2.4 Chebyshev Polynomials 2.4.1 Basic Formulas 2.4.2 Differentiation 2.5 Jacobi Polynomials 2.6 Approximation in Unbounded Domains 2.6.1 Laguerre Polynomials and Laguerre Functions 2.6.2 Hermite Polynomials and Hermite Functions 2.7 Mappings for Unbounded Domains 2.7.1 Semi-Infinite Intervals 2.7.2 The Real Line 2.8 Tensor-Product Expansions 2.8.1 Multidimensional Mapping 2.9 Expansions on Triangles and Related Domains 2.9.1 Collapsed Coordinates and Warped Tensor-Product Expansions 2.9.2 Non-Tensor-Product Expansions 2.9.3 Mappings 3. Basic Approaches to Constructing Spectral Methods 3.1 Burgers Equation 3.2 Strong and Weak Formulations of Differential Equations 3.3 Spectral Approximation of the Burgers Equation 3.3.1 Fourier Galerkin 3.3.2 Fourier Collocation 3.3.3 Chebyshev Tau 3.3.4 Chebyshev Collocation 3.3.5 Legendre G-NI 3.4 Convolution Sums 3.4.1 Transform Methods and Pseudospectral Methods 3.4.2 Aliasing Removal by Padding or Truncation 3.4.3 Aliasing Removal by Phase Shifts 3.4.4 Aliasing Removal for Orthogonal Polynomials 3.5 Relation Between Collocation, G-NI and Pseudospectral Methods 3.6 Conservation Forms 3.7 Scalar Hyperbolic Problems 3.7.1 Enforcement of Boundary Conditions 3.7.2 Numerical Examples 3.8 Matrix Construction for Galerkin and G-NI Methods 3.8.1 Matrix Elements 3.8.2 An Example of Algebraic Equivalence between G-NI and Collocation Methods 3.9 Polar Coordinates 3.10 Aliasing Effects 4. Algebraic Systems and Solution Techniques 4.1 Ad-hoc Direct Methods 4.1.1 Fourier Approximations 4.1.2 Chebyshev Tau Approximations 4.1.3 Galerkin Approximations 4.1.4 Schur Decomposition and Matrix Diagonalization 4.2 Direct Methods 4.2.1 Tensor Products of Matrices 4.2.2 Multidimensional Stiffness and Mass Matrices 4.2.3 Gaussian Elimination Techniques 4.3 Eigen-Analysis of Spectral Derivative Matrices 4.3.1 Second-Derivative Matrices 4.3.2 First-Derivative Matrices 4.3.3 Advection-Diffusion Matrices 4.4 Preconditioning 4.4.1 Fundamentals of Iterative Methods for Spectral Discretizations 4.4.2 Low-Order Preconditioning of Model Spectral Operators in One Dimension 4.4.3 Low-Order Preconditioning in Several Dimensions 4.4.4 Spectral Preconditioning 4.5 Descent and Krylov Iterative Methods for Spectral Equations 4.5.1 Multidimensional Matrix-Vector Multiplication 4.5.2 Iterative Methods 4.6 Spectral Multigrid Methods 4.6.1 One-Dimensional Fourier Multigrid Model Problem 4.6.2 General Spectral Multigrid Methods 4.7 Numerical Examples of Direct and Iterative Methods 4.7.1 Fourier Collocation Discretizations 4.7.2 Chebyshev Collocation Discretizations 4.7.3 Legendre G-NI Discretizations 4.7.4 Preconditioners for Legendre G-NI Matrices 4.8 Interlude 5. Polynomial Approximation Theory 5.1 Fourier Approximation 5.1.1 Inverse Inequalities for Trigonometric Polynomials 5.1.2 Estimates for the Truncation and Best Approximation Errors 5.1.3 Estimates for the Interpolation Error 5.2 Sturm-Liouville Expansions 5.2.1 Regular Sturm-Liouville Problems 5.2.2 Singular Sturm-Liouville Problems 5.3 Discrete Norms 5.4 Legendre Approximations 5.4.1 Inverse Inequalities for Algebraic Polynomials 5.4.2 Estimates for the Truncation and Best Approximation Errors 5.4.3 Estimates for the Interpolation Error 5.4.4 Scaled Estimates 5.5 Chebyshev Approximations 5.5.1 Inverse Inequalities for Polynomials 5.5.2 Estimates for the Truncation and Best Approximation Errors 5.5.3 Estimates for the Interpolation Error 5.6 Proofs of Some Approximation Results 5.7 Other Polynomial Approximations 5.7.1 Jacobi Polynomials 5.7.2 Laguerre and Hermite Polynomials 5.8 Approximation in Cartesian-Product Domains 5.8.1 Fourier Approximations 5.8.2 Legendre Approximations 5.8.3 Mapped Operators and Scaled Estimates 5.8.4 Chebyshev and Other Jacobi Approximations 5.8.5 Blended Trigonometric and Algebraic Approximations 5.9 Approximation in Triangles and Related Domains 6. Theory of Stability and Convergence 6.1 Three Elementary Examples Revisited 6.1.1 A Fourier Galerkin Method for the Wave Equation 6.1.2 A Chebyshev Collocation Method for the Heat Equation 6.1.3 A Legendre Tau Method for the Poisson Equation 6.2 Towards a General Theory 6.3 General Formulation of Spectral Approximations to Linear Steady Problems 6.4 Galerkin, Collocation, G-NI and Tau Methods 6.4.1 Galerkin Methods 6.4.2 Collocation Methods 6.4.3 G-NI Methods 6.4.4 Tau Methods 6.5 General Formulation of Spectral Approximations to Linear Evolution Problems 6.5.1 Conditions for Stability and Convergence: The Parabolic Case 6.5.2 Conditions for Stability and Convergence: The Hyperbolic Case 6.6 The Error Equation 7. Analysis of Model Boundary-Value Problems 7.1 The Poisson Equation 7.1.1 Legendre Methods 7.1.2 Chebyshev Methods 7.1.3 Other Boundary-Value Problems 7.2 Singularly Perturbed Elliptic Equations 7.2.1 Stabilization of Spectral Methods 7.3 The Eigenvalues of Some Spectral Operators 7.3.1 The Discrete Eigenvalues for Lu = −uxx 7.3.2 The Discrete Eigenvalues for Lu = −νuxx + βux 7.3.3 The Discrete Eigenvalues for Lu = ux 7.4 The Preconditioning of Spectral Operators 7.5 The Heat Equation 7.6 Linear Hyperbolic Equations 7.6.1 Periodic Boundary Conditions 7.6.2 Nonperiodic Boundary Conditions 7.6.3 The Resolution of the Gibbs Phenomenon 7.6.4 Spectral Accuracy for Non-Smooth Solutions 7.7 Scalar Conservation Laws 7.8 The Steady Burgers Equation Appendix A. Basic Mathematical Concepts A.1 Hilbert and Banach Spaces A.2 The Cauchy-Schwarz Inequality A.3 Linear Operators Between Banach Spaces A.4 The Fr´echet Derivative of an Operator A.5 The Lax-Milgram Theorem A.6 Dense Subspace of a Normed Space A.7 The Spaces Cm(Ω), m ≥ 0 A.8 Functions of Bounded Variation and the Riemann(-Stieltjes) Integral A.9 The Lebesgue Integral and Lp-Spaces A.10 Infinitely Differentiable Functions and Distributions A.11 Sobolev Spaces and Sobolev Norms A.12 The Sobolev Inequality A.13 The Poincar´e Inequality A.14 The Hardy Inequality A.15 The Gronwall Lemma Appendix B. Fast Fourier Transforms Appendix C. Iterative Methods for Linear Systems C.1 A Gentle Approach to Iterative Methods C.2 Descent Methods for Symmetric Problems C.3 Krylov Methods for Nonsymmetric Problems Appendix D. Time Discretizations D.1 Notation and Stability Definitions D.2 Standard ODE Methods D.2.1 Leap Frog Method D.2.2 Adams-Bashforth Methods D.2.3 Adams-Moulton Methods D.2.4 Backwards-Difference Formulas D.2.5 Runge-Kutta Methods D.3 Integrating Factors D.4 Low-Storage Schemes References Index

Note: Erscheinungsjahr in Vorlageform:1995

Note: CONTENTS Abstract Preface Introduction Equipment Data collection and reduction Data analysis and interpretation Point penetration resistance and material properties Lithologic and temperature profiles Casing penetration resistance Applications Shallow foundations Deep foundations Foundations in permafrost Correlations with laboratory tests Conclusions Selected bibliography Appendix A: Description of penetrometer and test procedures Appendix B: Point resistance and skin friction data

Note: CONTENT Abstract Preface Introduction Ice accretion parameters Physics of ice accretion Interaction between water droplets and flow field Time dependence in droplet trajectories Thermodynamic processes at the surface of the object Time dependence in the thermodynamics Time dependence of lesser order Numerical ice accretion model Major subroutines Options for droplet size variations Option for the helicopter rotor case Results Conclusions and future studies Literature cited Appendix A: Computer program

Note: Bremen, Univ., Diss., 2007

Note: Jean-Pierre Burg and Mary Ford: Orogeny through time: an overview / Geological Society, London, Special Publications, 121:1-17, doi:10.1144/GSL.SP.1997.121.01.01 --- Giorgio Ranalli: Rheology of the lithosphere in space and time / Geological Society, London, Special Publications, 121:19-37, doi:10.1144/GSL.SP.1997.121.01.02 --- P. G. Thomas, P. Allemand, and N. Mangold: Rheology of planetary lithospheres: a review from impact cratering mechanics / Geological Society, London, Special Publications, 121:39-62, doi:10.1144/GSL.SP.1997.121.01.03 --- P. Choukroune, J. N. Ludden, D. Chardon, A. J. Calvert, and H. Bouhallier: Archaean crustal growth and tectonic processes: a comparison of the Superior Province, Canada and the Dharwar Craton, India / Geological Society, London, Special Publications, 121:63-98, doi:10.1144/GSL.SP.1997.121.01.04 --- M. G. O’Dea, G. S. Lister, T. Maccready, P. G. Betts, N. H. S. Oliver, K. S. Pound, W. Huang, R. K. Valenta, N. H. S. Oliver, and R. K. Valenta: Geodynamic evolution of the Proterozoic Mount Isa terrain / Geological Society, London, Special Publications, 121:99-122, doi:10.1144/GSL.SP.1997.121.01.05 --- A. G. Milnes, O. P. Wennberg, Ø. Skår, and A. G. Koestler: Contraction, extension and timing in the South Norwegian Caledonides: the Sognefjord transect / Geological Society, London, Special Publications, 121:123-148, doi:10.1144/GSL.SP.1997.121.01.06 --- David R. Gray: Tectonics of the southeastern Australian Lachlan Fold Belt: structural and thermal aspects / Geological Society, London, Special Publications, 121:149-177, doi:10.1144/GSL.SP.1997.121.01.07 --- P. Rey, J.-P. Burg, and M. Casey: The Scandinavian Caledonides and their relationship to the Variscan belt / Geological Society, London, Special Publications, 121:179-200, doi:10.1144/GSL.SP.1997.121.01.08 --- V. N. Puchkov: Structure and geodynamics of the Uralian orogen / Geological Society, London, Special Publications, 121:201-236, doi:10.1144/GSL.SP.1997.121.01.09 --- Simon Lamb, Leonore Hoke, Lorcan Kennan, and John Dewey: Cenozoic evolution of the Central Andes in Bolivia and northern Chile / Geological Society, London, Special Publications, 121:237-264, doi:10.1144/GSL.SP.1997.121.01.10

Note: P. D. Jackson, D. G. Gunn, R. C. Flint, D. Beamish, P. I. Meldrum, M. A. Lovell, P. K. Harvey, and A. Peyton: A non-contacting resistivity imaging method for characterizing whole round core while in its liner / Geological Society, London, Special Publications, 122:1-10, doi:10.1144/GSL.SP.1997.122.01.01 --- D. J. Prosser, A. Hurst, and M. R. Wilson: One-man-operable probe permeameters / Geological Society, London, Special Publications, 122:11-18, doi:10.1144/GSL.SP.1997.122.01.02 --- S. R. McDougall, A. B. Dixit, and K. S. Sorbie: Network analogues of wettability at the pore scale / Geological Society, London, Special Publications, 122:19-35, doi:10.1144/GSL.SP.1997.122.01.03 --- S. Pugliese and N. Petford: Pore-structure visualization in microdioritic enclaves / Geological Society, London, Special Publications, 122:37-46, doi:10.1144/GSL.SP.1997.122.01.04 --- Paul B. Basan, Ben D. Lowden, Peter R. Whattler, and John J. Attard: Pore-size data in petrophysics: a perspective on the measurement of pore geometry / Geological Society, London, Special Publications, 122:47-67, doi:10.1144/GSL.SP.1997.122.01.05 --- M. S. King, A. Shakeel, and N. A. Chaudhry: Acoustic wave propagation and permeability in sandstones with systems of aligned cracks / Geological Society, London, Special Publications, 122:69-85, doi:10.1144/GSL.SP.1997.122.01.06 --- Shiyu Xu, Just Doorenbos, Sue Raikes, and Roy White: A simple but powerful model for simulating elastic wave velocities in clastic Silicate rocks / Geological Society, London, Special Publications, 122:87-105, doi:10.1144/GSL.SP.1997.122.01.07 --- T. Apuani, M. S. King, C. Butenuth, and M. H. De Freitas: Measurements of the relationship between Sonic wave velocities and tensile strength in Anisotropic rock / Geological Society, London, Special Publications, 122:107-119, doi:10.1144/GSL.SP.1997.122.01.08 --- C. McCann, J. Sothcott, and S. B. Assefa: Prediction of petrophysical properties from seismic quality factor measurements / Geological Society, London, Special Publications, 122:121-130, doi:10.1144/GSL.SP.1997.122.01.09 --- Y. F. Sun and D. Goldberg: Estimation of aspect-ratio changes with pressure from seismic velocities / Geological Society, London, Special Publications, 122:131-139, doi:10.1144/GSL.SP.1997.122.01.10 --- P. K. Harvey, M. A. Lovell, J. C. Lofts, P. A. Pezard, and J. F. Bristow: Petrophysical estimation from downhole Mineralogy logs / Geological Society, London, Special Publications, 122:141-157, doi:10.1144/GSL.SP.1997.122.01.11 --- Paul F. Worthington: Petrophysical estimation of permeability as a function of scale / Geological Society, London, Special Publications, 122:159-168, doi:10.1144/GSL.SP.1997.122.01.12 --- C. A. Gonçalves, P. K. Harvey, and M. A. Lovell: Prediction of petrophysical parameter logs using a multilayer backpropagation neural network / Geological Society, London, Special Publications, 122:169-180, doi:10.1144/GSL.SP.1997.122.01.13 --- Brian P. Moss: The partitioning of petrophysical data: a review / Geological Society, London, Special Publications, 122:181-252, doi:10.1144/GSL.SP.1997.122.01.14 --- A. Revil, P. A. Pezard, and M. Darot: Electrical conductivity, spontaneous potential and ionic diffusion in porous media / Geological Society, London, Special Publications, 122:253-275, doi:10.1144/GSL.SP.1997.122.01.15 --- Brígida Ramati P. Da Rocha and Tarek M. Habashy: Fractal Geometry, porosity and complex resistivity: from rough pore interfaces to hand specimens / Geological Society, London, Special Publications, 122:277-286, doi:10.1144/GSL.SP.1997.122.01.16 --- Brígida Ramati P. Da Rocha and Tarek M. Habashy: Fractal Geometry, porosity and complex resistivity: from hand specimen to field data / Geological Society, London, Special Publications, 122:287-297, doi:10.1144/GSL.SP.1997.122.01.17 --- M. Ben Clennell: Tortuosity: a guide through the maze / Geological Society, London, Special Publications, 122:299-344, doi:10.1144/GSL.SP.1997.122.01.18 --- J. C. Lofts, J. Bedford, H. Boulton, J. A. van Doorn, and P. Jeffreys: Feature recognition and the interpretation of images acquired from horizontal wellbores / Geological Society, London, Special Publications, 122:345-365, doi:10.1144/GSL.SP.1997.122.01.19 --- D. Goldberg and Y. F. Sun: Scattering attenuation as a function of depth in the Upper Oceanic Crust / Geological Society, London, Special Publications, 122:367-375, doi:10.1144/GSL.SP.1997.122.01.20 --- N. Passas, C. Butenuth, and M. H. De Freitas: An application of the Moiré Method to a study of local strains during rock failure in tension / Geological Society, London, Special Publications, 122:377-388, doi:10.1144/GSL.SP.1997.122.01.21

Note: CONTENTS Abstract Preface Introduction Historical literature review Relationships between midwinter temperature and freezing season Mean freezing index 4 Average January air temperature U.S. comparative study Correlation of AJAT and duration of freezing season Relationships between average January air temperature and snow conditions U.S. relationships German relationships Mapping of average January air temperatures Analysis of observed AJ AT data Development of detailed maps Application and discussion Examples Influence of vegetation Literature cited Appendix A. interpretation of snow cover by satellite Appendix B. Contour, station elevation and Aj AT maps for East and West Germany

Note: CONTENTS Abstract Preface List of symbols Summary Introduction Recent ice research in Denmark General Structures with vertical faces Wedges with inclined faces Recording of the strength of natural ice Artificial ice — properties and manufacture General Composition of material Order of mixing ingredients Rupture theory Theoretical approach Initial contact Forces acting on wedge Maximum force and actual force Principal stress equations 8Dynamic equations Intermittent nature of force Theoretical curve Verifiction of theory by model tests Description Width of rupture channel Plotting of results Engineering application Comparison with Korahavin’s results Recommendations for future research References Appendix A

Note: CONTENTS Abstract Preface Foreword Introduction Terminology Forces on individual cutters Torque force and tool force Forces on the rotor axis Tractive thrust and down thrust Alternative tool force formulations Vehicle traction Power/weight ratio Weight and balance Force, torque, speed and power Specific energy Efficiency and performance index Power density Power requirements for ejection of cuttings Hydrodynamic resistance in underwater cutting

Note: CONTENTS Abstract Preface Introduction Theory Limiting span of an arch Flow of pack ice through converging channels Stoppage of flow Applications St. Lawrence Amundsen Gulf Bering Strait Conclusion References

Note: CONTENTS Abstract Preface Conversion factors: U.S. customary to metric units of measurements Introduction The freezing proeess Icing observations Geographical distribution of icing and contributing meteorological conditions Extreme icing conditions Extremes of icc accumulation Prediction of icing occurrences Prediction of icing intensity and rate Control methods Measurement of icing rate Conclusions and recommendations Literature cited Appendix A. Maps of icing occurrence and rate Appendix B. Data for computing ship icing rates Appendix C. Estimation of ice accumulation

Note: CONTENTS Page Abstract Preface Summary Nomenclature Introduction Water movement through snow Vertical percolation Lateral flow Flat roofs Sloping roofs Rainfall intensity-duration effects Miscellaneous effects Radial flow to drains Flow along gutters on snow-covered roofs Snow structure Basal layer Conclusion Literature citied Appendix A. Computer program calculating roof loads from rain-on-snow

Note: CONTENTS Abstract Preface Summary Introduction Ice thickness and quantity Chronology of the 1975 Chena River breakup Checkpoint 3 Checkpoint 4 Checkpoint 5 Checkpoint 6 Checkpoint A Checkpoint 7 Checkpoint C Checkpoint D Checkpoint 8 Checkpoint 10 Checkpoint 11 Checkpoint 12 Spring flooding and debris The 1976 Chena River breakup Breakup in other years Summary and conclusions Appendix A. Sequential photographs of 1975 breakup at checkpoints 3-12

Note: CONTENTS Abstract Preface Nomenclature Conversion factors: U.S. customary to metric units of measurement Introduction Description of experiment Apparatus Test procedure Experimental results Summary of experimental tests Influence of coolant water flow rates Approximation of the rate of melting Comparison of experimental and computed results Comparative analysis of the model sinks Approximation of the heat transfer coefficient Conclusions and recommendations Literature cited Appendix A. Experimental data, heat sink Model II Appendix B. Heat sink scaling and similarity relationships Appendix C. Derivation of the relationships for the heat transfer coefficient, Reynolds and Nusselt numbers Appendix D. Determination of freezing rates and refrigeration loads Appendix E. Approximation of stresses in the heat sink tank

Note: Oslo, Univ., Diss., 2007