https://ntrs.nasa.gov/api/citations/20010050996/downloads/20010050996.pdf

Equivalent-Continuum Modeling of Nano-Structured Materials

https://www.tuc.gr/fileadmin/users_data/elci/Kalaitzakis/J.41.pdf

A roadmap towards intelligent net zero- and positive-energy buildings

http://gec.di.uminho.pt/DISCIP/MMNT/ModSim-0809/ArtigosAlunos/ModSimNanoMaterials_Odegard2005.pdf

Computational materials: Multi-scale modeling and simulation of nanostructured materials

Preface-Acknowledgements- I "Invariant Variational Problems" II- Invariance and Conservation Laws in the Twentieth Century The Inception and Reception of the Noether Theorems - 1 The Inception of the Noether Theorems -2 The Noether Theorems-3 The Noether Theorems as Seen by Contemporaries and by Historians of Science-4 From Bessel-Hagen to Hill, 1921-1951-5 The Reception of Noether's First Theorem after 1950-6 The Reception of Noether's Second Theorem after 1950-7 After 1970-Genuine Generalizations-Conclusion-Appendix I Postcard from Noether to Klein-Appendix II Letter from Noether to Klein -Appendix III Letter from Klein to Pauli-Appendix IV Letter from Noether to Einstein-Lectures at the Mathematical Society of Gottingen-References-Index

The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century

https://escholarship.org/content/qt5zt6c00q/qt5zt6c00q.pdf?t=q7nfab

State-of-the-art on research and applications of machine learning in the building life cycle

General Introduction. Problem Definition and Development: Introduction. Principle of Minimum Potential Energy. Elements of the Calculus of Variations. Derivation of the Plane Stress Problem. Rayleigh-Ritz Variational Solution Technique. Physically Interpretable Displacement Polynomials: Strain Gradient Notation: Introduction. Strain Gradient Notation. Strain Gradient Representation of Discrete Structures. Strain Transformations. A-Priori Error Analysis Procedures: Introduction. The Development of Strain Gradient Based Finite Elements. Four Node Quadrilateral Element. Six Node Linear Strain Element. Eight and Nine Node Elements. Shear Locking and Aspect Ratio Stiffening. The Strain Gradient Reformation of the Finite Differences Method: Introduction. Elements of the Finite Difference Method. Finite Difference Boundary Condition Models. Extensions to the Finite Difference Method. A-Posteriori Error Analysis Procedures: Introduction. The Zienkiewicz/Zhu Error Estimation Procedure. Error Estimation Based on Finite Difference Smoothing. Point-Wise Error Estimates. Super-Convergence of the Augmented Finite Element Results. Adaptive Refinement of Finite Difference Models. Subject Index.

A Unified Approach to the Finite Element Method and Error Analysis Procedures

Procedures for extending the useful scope of the finite difference method in solid mechanics applications are presented. The improvements centre around the introduction of the physical nature of the deformations into the equations used to formulate the approximate solution. This is accomplished by evaluating the coefficients of Taylor series expansions for the displacement approximations in terms of rigid body motions, strains and derivatives of strains. This procedure is demonstrated with plane stress applications. The ability to interpret the derivative approximations physically allows the fictitious nodes typical of the finite difference method to be rationally incorporated into the model as a means of enforcing traction boundary conditions. An example problem is solved using both regular and irregular meshes. The displacements and stresses compare well with finite element solutions.

A new approach to boundary modelling for finite difference applications in solid mechanics

The causes of shear locking and other discretization errors are analysed using a physically interpretable notation. This analysis provides insights that allow the errors due to shear locking to be removed either directly or indirectly. St. Venant's principle is incorporated into the stiffness matrix to directly eliminate shear locking from bending elements. Rules-of-thumb are suggested by the same analysis that will insure the absence of errors due to shear locking at the cost of additional degrees of freedom. It is also shown that aspect ratio stiffening in membrane elements is partially due to the same modelling error that produces shear locking. The source of parasitic shear is also identified and a direct procedure for eliminating it is given. A two node Timoshenko beam element and a four node membrane element are fully developed in symbolic form. The procedure is directly applicable to plate bending elements.

The identification and elimination of artificial stiffening errors in finite elements

A procedure for determining the equivalent continuum properties of a structure composed of repeated patterns of discrete elements with both displacement and rotational coordinates is presented. These nodal coordinates are transformed to rigid-body and strain gradient variables using a polynomial approximation. The maximum number of independent variables that may be retained is determined by applying a ranking procedure to the resulting transformation matrix. The possibility of introducing errors by requiring the analyst to supply the strain gradient terms directly is reduced by identifying the appropriate variables through the use of the polynomial expansion and the ranking procedure. Additional constraints may be imposed in this analysis. The equivalent continuum parameters result when a further transformation to the appropriate kinematic variables is applied and the strain energy expression is reduced to these variables. Three-dimensional beamand plate-like structures are treated. The results correspond to findings using other approaches.

Equivalent continuum representation of structures composed of repeated elements

The equivalent continuum properties of a structure composed of repeated patterns of discrete elements with both displacement and rotation coordinates are determined. These nodal coordinates are transformed to rigid body and strain gradient variables using a polynomial representation. The set of independent strain gradient variables is identified by inspection and depends on the geometry of the structure being modeled. The procedure is applied to six example problems, including two in which the effect of structural damage is analyzed.

John O. Dow

Papers

A Unified Approach to the Finite Element Method and Error Analysis Procedures

A new approach to boundary modelling for finite difference applications in solid mechanics

The identification and elimination of artificial stiffening errors in finite elements

Equivalent continuum representation of structures composed of repeated elements

Continuum Models of Space Station Structures