The Experts below are selected from a list of 2022 Experts worldwide ranked by ideXlab platform
W C Hassenpflug - One of the best experts on this subject based on the ideXlab platform.
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matrix Tensor Notation part ii skew and curved coordinates
Computers & Mathematics With Applications, 1995Co-Authors: W C HassenpflugAbstract:In Part I, a Notation called Matrix-Tensor Notation was introduced for rectilinear orthogonal coordinates. Part II discusses how the same Notation is equally efficient for vectors and 2nd order Tensors in skew bases and skew curved coordinates. Without considering coordinates first, vectors and 2nd order Tensors are described in skew bases, covariant and contravariant descriptions being distinguished similar to Tensor Notation. A consistent interpretation of deformation and strain in terms of skew bases is given. The use of the transpose symbol complicates the matrix algebra, but integrates it with Tensor algebra which makes it possible to interpret customary Tensor equations as relations in space with all the advantages that an isomorphism with Euclidean space has. It is demonstrated how different metrics can be assigned arbitrarily to define higher-dimensional orthogonal vectors in abstract higher-dimensional space. A consistent Notation is given to distinguish between subspace and subbase. To write higher order Tensors as matrices, or 2nd order Tensors as vectors, partial transpose is introduced, which is a formal stretching operation to write any higher order Tensor product operation as matrix-vector product. The description of the variable bases in generally curved coordinates is given in this Notation, together with the corresponding Notation for the partial differentiation rules. The vector operations grad, div and curl are discussed and an almost-physical Notation for the Christoffel symbols is given. However, the replacement of Tensor Notation for higher than second order Tensors by Matrix Tensor Notation is not feasible. A version of Matrix Tensor Notation that merges with the customary printed form is presented in an Appendix. Further applications are deferred to Part III, applying the new Notation to curved coordinates in function space, and the equations of mechanics projected on finite-dimensional subspaces.
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matrix Tensor Notation part i rectilinear orthogonal coordinates
Computers & Mathematics With Applications, 1993Co-Authors: W C HassenpflugAbstract:Abstract A Notation for vectors (first order Tensors) and Tensors (second order Tensors) in physical three-dimensional space is proposed that satisfies a number of requirements which are missing in the customary vector, matrix and Tensor Notations. It is designed particularly to distinguish between vectors and Tensors and their representation as vectors and matrices in different coordinate systems. This is achieved by the introduction of the base as a Tensor quantity in the fundamental equation for the relation between a Tensor and its representation as a matrix. The main purpose of the new Notation is that it can be used in the teaching situation, therefore, it conveys all the information explicitly in the symbols, and it can be used in handwriting. All different numerical quantities have different symbols so that in numerical and symbolic computer applications the symbols can be copied directly to similar computer names which provides for well chosen names of variables in a computer program. It is also shown that the same Notation is equally useful for vectors in abstract higher dimensional space and transformations and transforms in function space. Part I discusses the Notation for orthogonal coordinates, including Cartesian coordinates for which the Notation is very simple. Part II discusses how the same Notation is equally efficient for skew and curved coordinates, and how it is integrated with Tensor Notation for higher than second order Tensors.
L Chioncel - One of the best experts on this subject based on the ideXlab platform.
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dynamical mean field theory of the anderson hubbard model with local and nonlocal disorder in Tensor formulation
Physical Review B, 2021Co-Authors: Andreas Weh, Y Zhang, Andreas Ostlin, Hanna Terletska, D Bauernfeind, Kaming Tam, Hans Gerd Evertz, Krzysztof Byczuk, D Vollhardt, L ChioncelAbstract:To explore correlated electrons in the presence of local and nonlocal disorder, the Blackman-Esterling-Berk method for averaging over off-diagonal disorder is implemented into dynamical mean-field theory using Tensor Notation. The impurity model combining disorder and correlations is solved using the recently developed fork Tensor-product state solver, which allows one to calculate the single particle spectral functions on the real-frequency axis. In the absence of off-diagonal hopping, we establish exact bounds of the spectral function of the noninteracting Bethe lattice with coordination number $Z$. In the presence of interaction, the Mott insulating paramagnetic phase of the one-band Hubbard model is computed at zero temperature in alloys with site- and off-diagonal disorder. When the Hubbard $U$ parameter is increased, transitions from an alloy band insulator through a correlated metal into a Mott insulating phase are found to take place.
Andreas Weh - One of the best experts on this subject based on the ideXlab platform.
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dynamical mean field theory of the anderson hubbard model with local and nonlocal disorder in Tensor formulation
Physical Review B, 2021Co-Authors: Andreas Weh, Y Zhang, Andreas Ostlin, Hanna Terletska, D Bauernfeind, Kaming Tam, Hans Gerd Evertz, Krzysztof Byczuk, D Vollhardt, L ChioncelAbstract:To explore correlated electrons in the presence of local and nonlocal disorder, the Blackman-Esterling-Berk method for averaging over off-diagonal disorder is implemented into dynamical mean-field theory using Tensor Notation. The impurity model combining disorder and correlations is solved using the recently developed fork Tensor-product state solver, which allows one to calculate the single particle spectral functions on the real-frequency axis. In the absence of off-diagonal hopping, we establish exact bounds of the spectral function of the noninteracting Bethe lattice with coordination number $Z$. In the presence of interaction, the Mott insulating paramagnetic phase of the one-band Hubbard model is computed at zero temperature in alloys with site- and off-diagonal disorder. When the Hubbard $U$ parameter is increased, transitions from an alloy band insulator through a correlated metal into a Mott insulating phase are found to take place.
M Epstein - One of the best experts on this subject based on the ideXlab platform.
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cardiovascular solid mechanics cells tissues and organs
Applied Mechanics Reviews, 2002Co-Authors: Jay D Humphrey, M EpsteinAbstract:1. INTRODUCTION. 1.1 Historical Prelude, 1.2 Basic Cell Biology, 1.3 The Extracellular Matrix, 1.4 Soft Tissue Behavior, 1.5 Needs and General Approach, 1.6 Exercises, 1.7 References. 2. MATHEMATICAL PRELIMINARIES 2.1 A Direct Tensor Notation, 2.2 Cartesian Components, 2.3 Further Results in Tensor Calculus, 2.4 Orthogonal Curvilinear Components, 2.5 Matrix Methods, 2.6 Exercises, 2.7 References, 3. CONTINUUM MECHANICS 3.1 Kinematics, 3.2 Forces, Tractions and Stresses, 3.3 Balance Relations, 3.4 Constitutive Formulations, 3.5 Boundary and Initial Conditions, 3.6 Exercises, 3.7 References, 4. FINITE ELASTICITY 4.1 Introduction, 4.2 Incompressible Isotropic Elasticity, 4.3 Solutions in 3-D Incompressible Elasticity, 4.4 Compressible Isotropic Elasticity, 4.5 Membrane Hyperelasticity, 4.6 Exercises, 4.7 References 5. EXPERIMENTAL METHODS 5.1 General Philosophy, 5.2 Measurement of Strain, 5.3 Measurement of Applied Loads, 5.4 Testing Conditions, 5.5 Parameter Estimation and Statistics, 5.6 Exercises, 5.7 References 6. Finite Element Methods 6.1 Fundamental Equations, 6.2 Interpolation, Integration, and Solvers, 6.3 An Illustrative Formulation, 6.4 Inflation of a Membrane, 6.5 Inverse Finite Elements, 6.6 Exercises, 6.7 References PART II - VASCULAR MECHANICS 7. THE NORMAL ARTERIAL WALL 7.1 Structure and Function, 7.2 General Characteristics, 7.3 Constitutive Framework, 7.4 Experimental Methods, 7.5 Specific Constitutive Relations, 7.6 Stress Analyses, 7.7 Exercises, 7.8 References 8. VASCULAR DISORDERS 8.1 Hypertension, 8.2 Intracranial Aneurysms, 8.3 Atherosclerosis, 8.4 Aortic Aneurysms, 8.5 Additional Topics, 8.6 Exercises, 8.7 References 9. VASCULAR ADAPTATION 9.1 Mechanical Preliminaries, 9.2 Cellular Responses to Applied Loads, 9.3 Arterial Response to Hypertension, 9.4 Arterial Response to Altered Flow, 9.5 Vessel Response to Injury, 9.6 Veins as Arterial Grafts, 9.7 Aging, 9.8 Exercises, 9.9 References PART III CARDIAC MECHANICS 10. THE NORMAL HEART 10.1 Structure and Function, 10.2 General Characteristics, 10.3 Constitutive Framework, 10.4 Constitutive Relations, 10.5 Stress Analyses, 10.6 Exercises, 10.7 References 11. EPILOGUE APPENDICES I. Nomenclature, Abbreviations, and Conversions II. Results for Curvilinear Coordinates III. Material Frame Indifference 11. CARDIAC DISORDERS 11.1 Ischemia 11.2 Volume Overload 11.3 Hypertrophy 11.4 Cardiac Aneurysms 11.5 Additional Topics
Adejumobi Mudathir - One of the best experts on this subject based on the ideXlab platform.
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Tensors and their applications in mechanics
Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020Co-Authors: Adejumobi MudathirAbstract:The Tensor theory is a branch of Multilinear Algebra that describes the relationship between sets of algebraic objects related to a vector space. Tensor theory together with Tensor analysis is usually known to be Tensor calculus. This thesis presents a formal category treatment on Tensor Notation, Tensor calculus, and differential manifold. The focus lies mainly on acquiring and understanding the basic concepts of Tensors and the operations over them. It looks at how Tensor is adapted to differential geometry and continuum mechanics. In particular, it focuses more attention on the application parts of mechanics such as; configuration and deformation, Tensor deformation, continuum kinematics, Gauss, and Stokes' theorem with their applications. Finally, it discusses the concept of surface forces and stress vector.The Tensor theory is a branch of Multilinear Algebra that describes the relationship between sets of algebraic objects related to a vector space. Tensor theory together with Tensor analysis is usually known to be Tensor calculus. This thesis presents a formal category treatment on Tensor Notation, Tensor calculus, and differential manifold. The focus lies mainly on acquiring and understanding the basic concepts of Tensors and the operations over them. It looks at how Tensor is adapted to differential geometry and continuum mechanics. In particular, it focuses more attention on the application parts of mechanics such as; configuration and deformation, Tensor deformation, continuum kinematics, Gauss, and Stokes' theorem with their applications. Finally, it discusses the concept of surface forces and stress vector.
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Tensors and their applications in mechanics
Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2020Co-Authors: Adejumobi MudathirAbstract:The Tensor theory is a branch of Multilinear Algebra that describes the relationship between sets of algebraic objects related to a vector space. Tensor theory together with Tensor analysis is usually known to be Tensor calculus. This thesis presents a formal category treatment on Tensor Notation, Tensor calculus, and differential manifold. The focus lies mainly on acquiring and understanding the basic concepts of Tensors and the operations over them. It looks at how Tensor is adapted to differential geometry and continuum mechanics. In particular, it focuses more attention on the application parts of mechanics such as; configuration and deformation, Tensor deformation, continuum kinematics, Gauss, and Stokes' theorem with their applications. Finally, it discusses the concept of surface forces and stress vector