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The Resource Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics, R.M. Brannon

Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics, R.M. Brannon

Label
Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics
Title
Rotation, reflection, and frame changes
Title remainder
orthogonal tensors in computational engineering mechanics
Statement of responsibility
R.M. Brannon
Title variation
Orthogonal tensors in computational engineering mechanics
Creator
Contributor
Author
Publisher
Subject
Language
eng
Summary
Whilst vast literature is available for the most common rotation-related tasks such as coordinate changes, most reference books tend to cover one or two methods, and resources for less-common tasks are scarce. Specialized research applications can be found in disparate journal articles, but a self-contained comprehensive review that covers both elementary and advanced concepts in a manner comprehensible to engineers is rare. Rotation, Reflection, and Frame Changes surveys a refreshingly broad range of rotation-related research that is routinely needed in engineering practice. By illustrating key concepts in computer source code, this book stands out as an unusually accessible guide for engineers and scientists in engineering mechanics
Member of
Biographical or historical data
For over 25 years, first as a principal researcher (and manager) at Sandia National Laboratories and more recently as an Associate Professor of Mechanical Engineering at the University of Utah and ASME fellow, Dr Brannon has developed practical engineering constitutive models for brittle and ductile material failure at high strain rates and large strains. Her research has investigated a wide range of materials including piezoelectric ceramics, armor ceramics, geological materials, energetic materials, and metals (usually for high-rate applications). Constitutive models she has developed are used in DoD and DOE production codes such as CTH and ALEGRA. Applications have included protective structures, underground structure integrity, electroactive power supplies, artificial hip implant rapid materials ranking, shock-induced vaporization, in vivo measurements of callus strains, and other numerous other problems in the applied sciences. Dr Brannon is particularly known for her monographs on tensor analysis, plasticity, code portability, code verification, and massive deformation kinematics in the material point method.
Cataloging source
CaBNVSL
http://library.link/vocab/creatorName
Brannon, R. M.
Dewey number
620.1
Illustrations
illustrations
Index
no index present
Intended audience
Graduate students and researchers principally in engineering and materials modelling. Mathematicians with an interest in theoretical mechanics and computational mechanics
LC call number
TA329
LC item number
.B737 2018eb
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorName
Institute of Physics (Great Britain)
Series statement
IOP expanding physics,
http://library.link/vocab/subjectName
  • Engineering mathematics
  • Mechanics, Applied
  • Mechanics, Analytic
  • Classical mechanics
  • SCIENCE / Mechanics / General
  • Engineering mathematics
  • Mechanics, Analytic
  • Mechanics, Applied
Target audience
adult
Label
Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics, R.M. Brannon
Instantiates
Publication
Note
"Version: 20180401"--Title page verso
Bibliography note
Includes bibliographical references
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 1. Introduction -- 2. Notation and tensor analysis essentials -- 2.1. Linear fractional transform -- 2.2. Visualizing rotations
  • 3. Orthogonal basis and coordinate transformations -- 3.1. Superimposed rotations -- 3.2. Basis rotations
  • 4. Rotation operations -- 4.1. Why apparent inconsistency in placement of the negative sign?
  • 5. Axis and angle of rotation -- 5.1. Euler-Rodrigues formula -- 5.2. Computing the rotation tensor given axis and angle -- 5.3. Corollary to the Euler-Rodrigues formula : existence of a preferred basis -- 5.4. Computing axis and angle given the rotation tensor
  • 6. Rotations contrasted with reflections -- 7. Quaternion representation of a rotation -- 7.1. Shoemake's form -- 7.2. Relationship between quaternion and axis/angle forms
  • 8. Dyadic form of an invertible linear operator -- 8.1. Special case : lab basis -- 8.2. Special case : dyadic form of a rotation operation -- 8.3. Constructing a rotation that will transform one specified vector to another specified vector -- 8.4. Constructing a rotation from knowledge of initial and final 'marker' locations in a body
  • 9. Sequential rotations -- 9.1. The distinction between fixed and follower axes -- 9.2. Roll, pitch, yaw : sequential rotations about fixed (laboratory) axes -- 9.3. Euler angles : sequential rotations about 'follower' axes
  • 10. Series expression for a rotation -- 10.1. Cayley transformations
  • 11. Spectrum of a rotation -- 12. Polar decomposition -- 12.1. Difficult definition of the deformation gradient -- 12.2. Intuitive definition of the deformation gradient -- 12.3. The Jacobian of the deformation -- 12.4. Invertibility of a deformation -- 12.5. Sequential deformations -- 12.6. Matrix analysis version of the polar-decomposition theorem -- 12.7. Polar decomposition--a hindsight intuitive interpretation -- 12.8. Variational interpretation of the polar decomposition -- 12.9. A more rigorous (classical) presentation of the polar-decomposition theorem -- 12.10. The 'fast' way to do a polar decomposition in two dimensions -- 12.11. Scaling properties of a polar decomposition -- 12.12. Classic method for obtaining a polar decomposition in 3D -- 12.13. Another iterative polar decomposition in 3D
  • 13. Strain measures -- 13.1. One-dimensional strain measures -- 13.2. Three-dimensional strain definitions
  • 14. Remapping, advecting, or interpolating rotations -- 14.1. Proposal 1 : Map and re-compute the polar decomposition -- 14.2. Proposal 2 : Discard the 'stretch' part of a mixed rotation -- 14.3. Proposal 3 : Advect the pseudo-rotation vectors -- 14.4. Proposal 4 : Mix the quaternions -- 14.5. Advection enhancement strategy #1 : solve the compatibility equations -- 14.6. Mixing enhancement strategy #2 : Lagrangian tracers
  • 15. Rates and other derivatives of rotation -- 15.1. The 'spin' tensor -- 15.2. The angular velocity vector -- 15.3. Angular velocity in terms of axis and angle of rotation -- 15.4. Derivatives of rotation with respect to angle and axis -- 15.5. Difference between vorticity and polar spin -- 15.6. The (commonly mis-stated) Gosiewski's theorem -- 15.7. Rates of sequential rotations -- 15.8. Rates of simultaneous rotations -- 15.9. Integration of rotation rates
  • 16. Variations of tensor-valued functions of scalars and vectors -- 16.1. A motivational example -- 16.2. A comment about rates of proper functions -- 16.3. The time rate of a principal function of a symmetric tensor -- 16.4. Time rate of the logarithmic strain
  • 17. Statistics of random orientation -- 17.1. Elementary probability and statistics refresher -- 17.2. Uniformly random unit vectors--the theory -- 17.3. Uniformly random unit vectors--alternative implementation -- 17.4. 'Centroidally random' unit vectors -- 17.5. 'Nautical' visualization of a rotation -- 17.6. Uniformly random rotations -- 17.7. A basic algorithm for generating a uniformly random rotation -- 17.8. Generalization to generate transversely isotropic orientation distributions -- 17.9. Alternative algorithm for generating a uniformly random rotation -- 17.10. Shoemake's algorithm for uniformly random rotations
  • 18. Introduction to material and tensor symmetries -- 18.1. Anisotropy classification via group theory -- 18.2. Quantifying and visualizing orientations
  • 19. Frame indifference -- 19.1. A 3D spring--who expected it would be this hard!? -- 19.2. Introduction to frame indifference -- 19.3. Kinematics changes under superimposed rigid motion -- 19.4. Mechanics principles frame change
  • 20. Tensor symmetry (not material symmetry) -- 20.1. What is isotropy of a tensor? -- 20.2. Isotropic second-order tensors in 3D space -- 20.3. Isotropic second-order tensors in 2D space -- 20.4. Isotropic fourth-order tensors in 3D -- 20.5. The isotropic part of a fourth-order tensor -- 20.6. Tensor transverse isotropy -- 20.7. Material transverse isotropy
  • 21. Scalars and invariants -- 22. PMFI for incremental constitutive models -- 22.1. A frame-indifferent spring rate equation -- 22.2. The PMFI in general -- 22.3. PMFI in rate forms of the constitutive equations -- 22.4. Co-rotational rates (convected, Jaumann, polar, etc) -- 22.5. Lie derivatives and reference configurations -- 22.6. Frame indifference is only an essential (not final) step
  • 23. Rigid-body mechanics -- 23.1. Rate of rotation -- 23.2. The slope-intercept of rigid motion -- 23.3. The point-slope description of rigid motion -- 23.4. Velocity and angular velocity for rigid motion -- 23.5. Time rate of a vector embedded in a rigid body -- 23.6. Acceleration for rigid motion -- 23.7. Important properties of a rigid body -- 23.8. Linear momentum of a rigid body -- 23.9. Angular momentum of a rigid body -- 23.10. Kinetic energy of a rigid body -- 23.11. Newton's equation (balance of linear momentum) -- 23.12. Euler's equation (balance of angular momentum)
  • 24. Pseudo-body force for spinning problems -- 24.1. Kinematics of superimposed rotation (general analysis) -- 24.2. Fiducial body force for superimposed rigid motion
  • 25. Computer graphics visualization -- 25.1. Orientation of the body -- 25.2. Mapping from the body to the screen -- 25.3. Mapping from the screen to the virtual visible surface -- 25.4. Changing the screen image of a body
  • 26. Voigt and Mandel components -- 26.1. An introductory 3D example -- 26.2. Voigt components (inefficient and error prone!) -- 26.3. Mandel components (nice!) -- 26.4. Voigt components of fourth-order minor-symmetric tensors -- 26.5. Mandel components of fourth-order minor-symmetric tensors -- 26.6. Mandel components of fourth-order general tensors -- 26.7. Fourth-order linear transformations -- 26.8. Spectral analysis of fourth-order tensors
  • 27. Higher-order rotations -- 27.1. Rotators : fourth-order rotations in Mandel form -- 27.2. Fourth-order 'focused identity' (projection) tensors -- 27.3. Focused rotations -- 27.4. Components of focused identities and elided projectors -- 27.5. Single-plane fourth-order rotations -- 27.6. Preferred basis for single-plane rotation -- 27.7. Double-plane fourth-order rotations -- 27.8. Multi-plane fourth-order rotations -- 28. Closing remarks
Control code
1034808900
Dimensions
unknown
Extent
1 online resource (1 volume (various pagings))
File format
multiple file formats
Form of item
online
Isbn
9780750314534
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1016/978-0-7503-1454-1
Other physical details
color illustrations.
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1034808900
Label
Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics, R.M. Brannon
Publication
Note
"Version: 20180401"--Title page verso
Bibliography note
Includes bibliographical references
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • 1. Introduction -- 2. Notation and tensor analysis essentials -- 2.1. Linear fractional transform -- 2.2. Visualizing rotations
  • 3. Orthogonal basis and coordinate transformations -- 3.1. Superimposed rotations -- 3.2. Basis rotations
  • 4. Rotation operations -- 4.1. Why apparent inconsistency in placement of the negative sign?
  • 5. Axis and angle of rotation -- 5.1. Euler-Rodrigues formula -- 5.2. Computing the rotation tensor given axis and angle -- 5.3. Corollary to the Euler-Rodrigues formula : existence of a preferred basis -- 5.4. Computing axis and angle given the rotation tensor
  • 6. Rotations contrasted with reflections -- 7. Quaternion representation of a rotation -- 7.1. Shoemake's form -- 7.2. Relationship between quaternion and axis/angle forms
  • 8. Dyadic form of an invertible linear operator -- 8.1. Special case : lab basis -- 8.2. Special case : dyadic form of a rotation operation -- 8.3. Constructing a rotation that will transform one specified vector to another specified vector -- 8.4. Constructing a rotation from knowledge of initial and final 'marker' locations in a body
  • 9. Sequential rotations -- 9.1. The distinction between fixed and follower axes -- 9.2. Roll, pitch, yaw : sequential rotations about fixed (laboratory) axes -- 9.3. Euler angles : sequential rotations about 'follower' axes
  • 10. Series expression for a rotation -- 10.1. Cayley transformations
  • 11. Spectrum of a rotation -- 12. Polar decomposition -- 12.1. Difficult definition of the deformation gradient -- 12.2. Intuitive definition of the deformation gradient -- 12.3. The Jacobian of the deformation -- 12.4. Invertibility of a deformation -- 12.5. Sequential deformations -- 12.6. Matrix analysis version of the polar-decomposition theorem -- 12.7. Polar decomposition--a hindsight intuitive interpretation -- 12.8. Variational interpretation of the polar decomposition -- 12.9. A more rigorous (classical) presentation of the polar-decomposition theorem -- 12.10. The 'fast' way to do a polar decomposition in two dimensions -- 12.11. Scaling properties of a polar decomposition -- 12.12. Classic method for obtaining a polar decomposition in 3D -- 12.13. Another iterative polar decomposition in 3D
  • 13. Strain measures -- 13.1. One-dimensional strain measures -- 13.2. Three-dimensional strain definitions
  • 14. Remapping, advecting, or interpolating rotations -- 14.1. Proposal 1 : Map and re-compute the polar decomposition -- 14.2. Proposal 2 : Discard the 'stretch' part of a mixed rotation -- 14.3. Proposal 3 : Advect the pseudo-rotation vectors -- 14.4. Proposal 4 : Mix the quaternions -- 14.5. Advection enhancement strategy #1 : solve the compatibility equations -- 14.6. Mixing enhancement strategy #2 : Lagrangian tracers
  • 15. Rates and other derivatives of rotation -- 15.1. The 'spin' tensor -- 15.2. The angular velocity vector -- 15.3. Angular velocity in terms of axis and angle of rotation -- 15.4. Derivatives of rotation with respect to angle and axis -- 15.5. Difference between vorticity and polar spin -- 15.6. The (commonly mis-stated) Gosiewski's theorem -- 15.7. Rates of sequential rotations -- 15.8. Rates of simultaneous rotations -- 15.9. Integration of rotation rates
  • 16. Variations of tensor-valued functions of scalars and vectors -- 16.1. A motivational example -- 16.2. A comment about rates of proper functions -- 16.3. The time rate of a principal function of a symmetric tensor -- 16.4. Time rate of the logarithmic strain
  • 17. Statistics of random orientation -- 17.1. Elementary probability and statistics refresher -- 17.2. Uniformly random unit vectors--the theory -- 17.3. Uniformly random unit vectors--alternative implementation -- 17.4. 'Centroidally random' unit vectors -- 17.5. 'Nautical' visualization of a rotation -- 17.6. Uniformly random rotations -- 17.7. A basic algorithm for generating a uniformly random rotation -- 17.8. Generalization to generate transversely isotropic orientation distributions -- 17.9. Alternative algorithm for generating a uniformly random rotation -- 17.10. Shoemake's algorithm for uniformly random rotations
  • 18. Introduction to material and tensor symmetries -- 18.1. Anisotropy classification via group theory -- 18.2. Quantifying and visualizing orientations
  • 19. Frame indifference -- 19.1. A 3D spring--who expected it would be this hard!? -- 19.2. Introduction to frame indifference -- 19.3. Kinematics changes under superimposed rigid motion -- 19.4. Mechanics principles frame change
  • 20. Tensor symmetry (not material symmetry) -- 20.1. What is isotropy of a tensor? -- 20.2. Isotropic second-order tensors in 3D space -- 20.3. Isotropic second-order tensors in 2D space -- 20.4. Isotropic fourth-order tensors in 3D -- 20.5. The isotropic part of a fourth-order tensor -- 20.6. Tensor transverse isotropy -- 20.7. Material transverse isotropy
  • 21. Scalars and invariants -- 22. PMFI for incremental constitutive models -- 22.1. A frame-indifferent spring rate equation -- 22.2. The PMFI in general -- 22.3. PMFI in rate forms of the constitutive equations -- 22.4. Co-rotational rates (convected, Jaumann, polar, etc) -- 22.5. Lie derivatives and reference configurations -- 22.6. Frame indifference is only an essential (not final) step
  • 23. Rigid-body mechanics -- 23.1. Rate of rotation -- 23.2. The slope-intercept of rigid motion -- 23.3. The point-slope description of rigid motion -- 23.4. Velocity and angular velocity for rigid motion -- 23.5. Time rate of a vector embedded in a rigid body -- 23.6. Acceleration for rigid motion -- 23.7. Important properties of a rigid body -- 23.8. Linear momentum of a rigid body -- 23.9. Angular momentum of a rigid body -- 23.10. Kinetic energy of a rigid body -- 23.11. Newton's equation (balance of linear momentum) -- 23.12. Euler's equation (balance of angular momentum)
  • 24. Pseudo-body force for spinning problems -- 24.1. Kinematics of superimposed rotation (general analysis) -- 24.2. Fiducial body force for superimposed rigid motion
  • 25. Computer graphics visualization -- 25.1. Orientation of the body -- 25.2. Mapping from the body to the screen -- 25.3. Mapping from the screen to the virtual visible surface -- 25.4. Changing the screen image of a body
  • 26. Voigt and Mandel components -- 26.1. An introductory 3D example -- 26.2. Voigt components (inefficient and error prone!) -- 26.3. Mandel components (nice!) -- 26.4. Voigt components of fourth-order minor-symmetric tensors -- 26.5. Mandel components of fourth-order minor-symmetric tensors -- 26.6. Mandel components of fourth-order general tensors -- 26.7. Fourth-order linear transformations -- 26.8. Spectral analysis of fourth-order tensors
  • 27. Higher-order rotations -- 27.1. Rotators : fourth-order rotations in Mandel form -- 27.2. Fourth-order 'focused identity' (projection) tensors -- 27.3. Focused rotations -- 27.4. Components of focused identities and elided projectors -- 27.5. Single-plane fourth-order rotations -- 27.6. Preferred basis for single-plane rotation -- 27.7. Double-plane fourth-order rotations -- 27.8. Multi-plane fourth-order rotations -- 28. Closing remarks
Control code
1034808900
Dimensions
unknown
Extent
1 online resource (1 volume (various pagings))
File format
multiple file formats
Form of item
online
Isbn
9780750314534
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1016/978-0-7503-1454-1
Other physical details
color illustrations.
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1034808900

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