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A new book and software for modeling of shells based on the Kirchhoff-Love theory

Available with source codes and intended for scientific and engineering research
Differential Geometry of Surfaces
The book "Computational Geometry of Surfaces and its Application to the Finite Element Analysis of Shells" by Olga Axenenko and Alexander Tsvelikh provides a theoretical introduction in differential geometry of surfaces, as well as mathematical fundamentals of the theory of thin shells.

The principal value of the book is a completely new numerical method that allows constructing the lines of principal curvature on any type of curved surface.

As an application of the developed method, we chose to create a new 9-node finite element and to test it on several practical shell structural problems which analytical solutions are known.

However, the method for constructing the lines of principal curvature might be used to solve other problems which require the knowlede of a surfaces geometry.

The surface geometry is essential for the definition of a shell structure.

Our objective in Chapter 1 of the book is to introduce the main geometrical concepts of the differential geometry of surfaces in great detail. We pay special attention to parameters of surface and explain their geometrical meaning.

The structure of Chapter 1 is as follows:
  1. Coordinate systems and basis vectors
  2. Vectors and tensors
  3. The metric tensor
  4. The first fundumental form of surface
  5. The second fundamental form of surface
  6. The discriminant tensor
  7. Principal directions and curvatures of a surface
  8. Covariant derivatives of vectors and tensors
  9. Orthogonal coordinate systems and orthogonal basis
  10. The physical components of vectors and tensors
  11. The coordinate systems of the lines of principal curvature
It was the intention of the authours that the book is rather self-contained. As a result the readers can study and/or use the book without a necessity to refer to other books or information sources, i.e. just a general mathematical and mechanical engineering background is required.

In Chapter 5 of the book we introduce a new method that allows construction of the lines of principal curvature on arbitrary complex curved surfaces. We explain in details the corresponding algorithm and its implementation.

The lines of principal curvature constructed by our algorithm at the center of the 9-node quadrilateral shell finite element can be chosen as the coordinate curves on the shell surface.
The coordinate system of the lines of principal curvature is particularly important because the lines of principle curvature represent a special case of an orthogonal coordinate system in which the espressions for the coordinate system parameters are geatly simplified.

Since in the coordinate system of the lines of principal curvature only the diagonal components of the metric tensor are non zeros, we can introduce the Lame coefficients.

Also, since only the diagonal components of the curvature tensor are non zeros, we can introduce main curvature values.

As a result, the governing equations of the theory of thin shells can be written in a very simple form.

Also, QUAD-SHELL algorithm allows computation of the Lame coefficients and the main curvatures, as well as their derivatives with respect to the coordinates of the lines of main curvature at any point on the finite element.

For example, we compute the values of these geometrical surface parameters at Gauss intergation points (of the Gauss quadrature used to compute the finite element area) and at points on the element where we calculate the deflection and the stress resultant forces.

QUAD-SHELL algorithm could be used as a basis for other methods of numerical analysis of problems for which the knowledge of the surface parameters is needed.

The material contained in Chapter 1 and 5 the book could make the base of a course of lectures for students.

The CD ROM, which is shipped with the book, contains the complete source code of the software program QUAD-SHELL written in C++ that can be immediately compiled and run.
The CD ROM also contains the User Manual as well as the input and output data files for
The book is available on Amazon as paperback or hardcover.
You can also order the book by contacting Computational Mechanics Australia Pty. Ltd. by e-mail on or

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