Read here about the finite element method (FEM) and its application to tensegrity research, both in structures and cellular mechanics.

Overview

The finite element method (FEM), known in its practical application as finite element analysis (FEA), is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc.

In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness.

FEM is applied in tensegrity structural research and in biological, cellular mechanical research.

Finite Element Analysis in Tensegrity Research

Arcaro reviewed seven form-finding methods for tensegrity structures: three kinematical methods and four statical methods.

Kinematic Methods as Applied to Tensegrity Structures

Kinematical methods that determine the geometry of a given tensegrity structure by maximising the lengths of the struts while keeping constant the given lengths of the cables. The characteristic of these methods is that the lengths of the cables are kept constant while the strut lengths are increased until a maximum is reached. Alternatively, the strut lengths may be kept constant while the cable lengths are decreased until they reach a minimum. This approach mimics the way in which tensegrity structures are built in practice, without explicitly requiring that the cables be put in a state of pre-tension.
The three kinematical methods are
1. analytical approach. Statical methods determine the possible equilibrium configurations of a tensegrity structure with given topology by using an abstract description of the tensegrity's geometry. Consider a simple structure consisting of cables arranged along the edges of a regular prism, plus a number of struts connecting the v vertices of the bottom polygon to corresponding vertices of the upper polygon. Depending on the value of v and the offset between vertices connected by a strut, there is a special rotation angle θ between the top and bottom polygons for which a tensegrity structure is obtained. A compact description of the geometry of this problem was introduced by Connelly and Terrell. Kenner used node equilibrium and symmetry arguments to find the configuration of the expandable octahedron, and other, more complex, spherical tensegrities with polyhedral geometries, i.e. the cuboctahedron and the icosidodecahedron, were also analysed using the same approach.
2. non-linear optimisation. This general method, proposed by Pellegrino, turns the form-finding of any tensegrity structure into a constrained minimisation problem. Starting from a system for which the element connectivity and nodal coordinates are known, one or more struts are elongated, maintaining fixed length ratios, until a configuration is reached in which their length is maximised.
3. pseudo-dynamic iteration, or dynamic relaxation. An application of a method already successfully used for membrane andcable net structures, put forward by Motro and Belkacem as a general form-finding method for tensegrity structures. For a structure in a given initial configuration and subject to given external forces the equilibrium of configuration is computed by integrating a set of fictitious dynamic equations involving a stiffness matrix, a mass matrix, a damping matrix, a vector of external forces, and vectors of acceleration, velocity and displacement from the initial configuration. Both the mass and damping matrices are taken to be diagonal, for simplicity, and the velocities and displacements are initially set to zero.

Static Methods as Applied to Tensegrity Structures

Statical methods determine the possible equilibrium configurations of a tensegrity structure with given topology by using an abstract description of the tensegrity's geometry. Consider a simple structure consisting of cables arranged along the edges of a regular prism, plus a number of struts connecting the v vertices of the bottom polygon to corresponding vertices of the upper polygon. Depending on the value of v and the offset between vertices connected by a strut, there is a special rotation angle θ between the top and bottom polygons for which a tensegrity structure is obtained. A compact description of the geometry of this problem was introduced by Connelly and Terrell.
The four statical methods are
4. analytical method. Establishes linear nodal equations of equilibrium in terms of so-called force densities and solve these equations for the nodal coordinates by analytically methods.
5. formulation of linear equations of equilibrium in terms of force densities. Same as (4) above, but it solves the equations for the nodal coordinates by setting up a force density matrix in method. The force density method for cable structures, first proposed by Linkwitz and Schek, uses a simple mathematical trick to transform the non-linear equilibrium equations of the nodes into a set of linear equations.
6. an energy minimisation. Based on an energy minimisation approach, which is shown to produce a matrix identical to the force-density matrix, and introduces the concept of super-stable tensegrities. The method is based on a basic principle in the analysisof the stability of structures, that for a given configuration to be stable, the total potential energy functional should be at a local minimum. To model this, Connelly stipulates a configuration of n ordered points in d-dimensional space. A tensegrity framework is then generated as the graph on the set of points, where each edge is designated as either a cable, a strut or a bar; cables cannot increase in length, struts cannot decrease in length and bars cannot change length. The method also stipulates stress states. Its application led to discovery of some previously unknown tensegrities, though it also generates models where the struts go through one another.
7. search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. The method searches for equilibrium configurations of a set of rigid bodies, i.e. the struts of the tensegrity structure, connected by cables whose lengths are to be determined. A reduced set of equilibrium equations for the struts are determined by virtual work, making use of symmetry conditions, and are then solved in symbolic form. The method was introduced by Sultan. Consider a tensegrity structure whose b elements consist of M cables and O struts. The struts are considered as a set of bilateralconstraints acting on the cable structure. Hence, a set of independent, generalised coordinates g is defined, which define the position and orientation of these struts. A state of self-stress for the structure is then considered, modelling the axial forces in generic cableelements in equilibrium with appropriate forces in the struts and zero external loads. A set of equilibrium equations relating the forces in the cables, but without showing explicitly the forces in the struts, can be obtained from virtual work.

Recommendations for Tensegrity Form Discovery Methods

Arcaro concluded:A. kinematical methods are best suited to obtaining only configuration details of structures that are already essentially knownB. the force density method (5) is best suited to searching for new configurations, but affords no control over the lengths of the elements of the structure. C. the reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations.

FEM in Cellular Mechanics

Shira Or-Tzadikarioa and Amit Gefen of Tel-Aviv University reported on confocal-based cell-specific finite element modeling extended to study variable cell shapes and intracellular structures. They focus on the adipocyte. The abstract: "This communication extends the recently reported cell-specific finite element (FE) method in Slomka and Gefen (2010) in which geometrically realistic FE cell models are created from confocal microscopy scans for large deformation analyses. The cell-specific FE method is extended here in the following aspects: (i) we demonstrate that cell-specific FE is versatile enough to deal with cells of substantially different geometrical shapes. The examples of an “elongated” pre-adipocyte and a “round” mature adipocyte are used to demonstrate this feature. (ii) We demonstrate that cell-specific FE can be used to analyze the mechanical behavior of cells that incorporate complex intracellular structures and are subjected to large deformations—again through the example of an adipocyte which contains a multitude of lipid droplets, each having a different size and shape. By demonstrating feasibility of inclusion of such inhomogeneities in the cytoplasm, the present work paves the way for modeling cellular organelles such as Golgi bodies, lysosomes and mitochondria in mechanically loaded cells using cell-specific FE." [2]

Quasi-Newton Finite Element Analysis

Vinicius Arcaro, Katalin Klinka, Dario Gasparini proposed in "Finite Element Analysis For Minimal Shape" using a quasi-Newton method that avoids the evaluation of the Hessian matrix as required in a Newton method. They published a text and computer source code.

This text describes a mathematical model for minimizing path length, surface area and volume using a line element, a triangle element and a tetrahedron element respectively. The elements can also be used together through the unifying concept of minimizing shape volume. The square of the relative change in the element’s volume is used to define an isovolumetric element. The source and executable computer codes of the algorithm are available from the first author's website, written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD. Link: http://www.arcaro.org/main.htm

Finite Element Analysis Software

Selected software products that enable FEM of tensegrity structures.

Rhinoceros Software and FEM

Rhinoceros Software Can Model Tensegrity Structures Using the Rhino-Membrane Plugin.

Rhino-Membrane is a plugin for Rhinoceros 4.0, a leading 3D modelling software solution. The plug-in enables a finite element approach method, or shape finding, within the graphical user interface provided by Rhino.
A video prepared by the Rhino-Membrane team, posted at their site http://www.membranes24.com , shows how to create tensegrity structures using this form finding process. The video shows construction of a 3 strut prism, and shows previously modelled 4 strut and 5 strut prisms. The video is hosted at http://www.youtube.com/watch?v=mGaKVU8ikPc and is embedded below. The structure is modelled with elastic truss elements and predefined prestress elements.

The steps in the video:
1. Create initial geometry using polylines using tensioned cables (elastic truss elements)
2. Connect points with polylines
3. Use form finding cables. The blue vertical cables have a defined prestress instead of elastic properties
4. Convert the compressive elements to truss.
5. Set the prestrain
6. Create point objects to fix polylines
7. Fix six spatial degrees of freedom for a statically determined boundary condition
8. Define the trusses under compression (elastic truss elements) as cables
9. Assign boundary conditions
10. Assign twist angle. Every prismatic tensegrity structure has a certain "twist angle" that is independent of its height.

Ansys Workbench

Ansys Workbench, a leading engineering simulation tool, can model tensegrity structures. Vitfc recently posted to an Ansys forum, "in order to study the efficency of tensegrity structures I am trying to write down the stiffness equations of such structures, to compare my equations with the answer of Finite Element Modelling software packages. I have done a really simply model in Ansys but I have some problems to model de behaviour of the cables in the structure. How can I model cables in Ansys Workbench? Should I choose the cables and change the properties of the material for anisotropic? or there is another way?" Below, a picture of the structure and the boundary conditions.

Modelling a TEnsegrity Structure in Ansys Workbench

## Finite Element Method

## Table of Contents

finite element method(FEM) and its application to tensegrity research, both in structures and cellular mechanics.## Overview

The finite element method (FEM), known in its practical application as finite element analysis (FEA), is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc.

In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness.

FEM is applied in tensegrity structural research and in biological, cellular mechanical research.

## Finite Element Analysis in Tensegrity Research

Arcaro reviewed seven form-finding methods for tensegrity structures: three kinematical methods and four statical methods.

## Kinematic Methods as Applied to Tensegrity Structures

Kinematical methods that determine the geometry of a given tensegrity structure by maximising the lengths of the struts while keeping constant the given lengths of the cables. The characteristic of these methods is that the lengths of the cables are kept constant while the strut lengths are increased until a maximum is reached. Alternatively, the strut lengths may be kept constant while the cable lengths are decreased until they reach a minimum. This approach mimics the way in which tensegrity structures are built in practice, without explicitly requiring that the cables be put in a state of pre-tension.

The three kinematical methods are

1.

analytical approach. Statical methods determine the possible equilibrium configurations of a tensegrity structure with given topology by using an abstract description of the tensegrity's geometry. Consider a simple structure consisting of cables arranged along the edges of a regular prism, plus a number of struts connecting the v vertices of the bottom polygon to corresponding vertices of the upper polygon. Depending on the value of v and the offset between vertices connected by a strut, there is a special rotation angle θ between the top and bottom polygons for which a tensegrity structure is obtained. A compact description of the geometry of this problem was introduced by Connelly and Terrell. Kenner used node equilibrium and symmetry arguments to find the configuration of the expandable octahedron, and other, more complex, spherical tensegrities with polyhedral geometries, i.e. the cuboctahedron and the icosidodecahedron, were also analysed using the same approach.2.

non-linear optimisation. This general method, proposed by Pellegrino, turns the form-finding of any tensegrity structure into a constrained minimisation problem. Starting from a system for which the element connectivity and nodal coordinates are known, one or more struts are elongated, maintaining fixed length ratios, until a configuration is reached in which their length is maximised.3.

pseudo-dynamic iteration, or dynamic relaxation. An application of a method already successfully used for membrane andcable net structures, put forward by Motro and Belkacem as a general form-finding method for tensegrity structures. For a structure in a given initial configuration and subject to given external forces the equilibrium of configuration is computed by integrating a set of fictitious dynamic equations involving a stiffness matrix, a mass matrix, a damping matrix, a vector of external forces, and vectors of acceleration, velocity and displacement from the initial configuration. Both the mass and damping matrices are taken to be diagonal, for simplicity, and the velocities and displacements are initially set to zero.## Static Methods as Applied to Tensegrity Structures

Statical methods determine the possible equilibrium configurations of a tensegrity structure with given topology by using an abstract description of the tensegrity's geometry. Consider a simple structure consisting of cables arranged along the edges of a regular prism, plus a number of struts connecting the v vertices of the bottom polygon to corresponding vertices of the upper polygon. Depending on the value of v and the offset between vertices connected by a strut, there is a special rotation angle θ between the top and bottom polygons for which a tensegrity structure is obtained. A compact description of the geometry of this problem was introduced by Connelly and Terrell.

The four statical methods are

4. analytical method. Establishes linear nodal equations of equilibrium in terms of so-called force densities and solve these equations for the nodal coordinates by analytically methods.

5. formulation of linear equations of equilibrium in terms of force densities. Same as (4) above, but it solves the equations for the nodal coordinates by setting up a force density matrix in method. The force density method for cable structures, first proposed by Linkwitz and Schek, uses a simple mathematical trick to transform the non-linear equilibrium equations of the nodes into a set of linear equations.

6. an energy minimisation. Based on an energy minimisation approach, which is shown to produce a matrix identical to the force-density matrix, and introduces the concept of super-stable tensegrities. The method is based on a basic principle in the analysisof the stability of structures, that for a given configuration to be stable, the total potential energy functional should be at a local minimum. To model this, Connelly stipulates a configuration of n ordered points in d-dimensional space. A tensegrity framework is then generated as the graph on the set of points, where each edge is designated as either a cable, a strut or a bar; cables cannot increase in length, struts cannot decrease in length and bars cannot change length. The method also stipulates stress states. Its application led to discovery of some previously unknown tensegrities, though it also generates models where the struts go through one another.

7. search for the equilibrium configurations of the struts of the structure connected by cables whose lengths are to be determined, using a reduced set of equilibrium equations. The method searches for equilibrium configurations of a set of rigid bodies, i.e. the struts of the tensegrity structure, connected by cables whose lengths are to be determined. A reduced set of equilibrium equations for the struts are determined by virtual work, making use of symmetry conditions, and are then solved in symbolic form. The method was introduced by Sultan. Consider a tensegrity structure whose b elements consist of M cables and O struts. The struts are considered as a set of bilateralconstraints acting on the cable structure. Hence, a set of independent, generalised coordinates g is defined, which define the position and orientation of these struts. A state of self-stress for the structure is then considered, modelling the axial forces in generic cableelements in equilibrium with appropriate forces in the struts and zero external loads. A set of equilibrium equations relating the forces in the cables, but without showing explicitly the forces in the struts, can be obtained from virtual work.

## Recommendations for Tensegrity Form Discovery Methods

Arcaro concluded:A. kinematical methods are best suited to obtaining only configuration details of structures that are already essentially knownB. the force density method (5) is best suited to searching for new configurations, but affords no control over the lengths of the elements of the structure. C. the reduced coordinates method offers a greater control on elements lengths, but requires more extensive symbolic manipulations.

## FEM in Cellular Mechanics

Shira Or-Tzadikarioa and Amit Gefen of Tel-Aviv University reported on confocal-based cell-specific

finite element modelingextended to study variable cell shapes and intracellular structures. They focus on the adipocyte. The abstract: "This communication extends the recently reported cell-specific finite element (FE) method in Slomka and Gefen (2010) in which geometrically realistic FE cell models are created from confocal microscopy scans for large deformation analyses. The cell-specific FE method is extended here in the following aspects: (i) we demonstrate that cell-specific FE is versatile enough to deal with cells of substantially different geometrical shapes. The examples of an “elongated” pre-adipocyte and a “round” mature adipocyte are used to demonstrate this feature. (ii) We demonstrate that cell-specific FE can be used to analyze the mechanical behavior of cells that incorporate complex intracellular structures and are subjected to large deformations—again through the example of an adipocyte which contains a multitude of lipid droplets, each having a different size and shape. By demonstrating feasibility of inclusion of such inhomogeneities in the cytoplasm, the present work paves the way for modeling cellular organelles such as Golgi bodies, lysosomes and mitochondria in mechanically loaded cells using cell-specific FE." [2]## Quasi-Newton Finite Element Analysis

Vinicius Arcaro, Katalin Klinka, Dario Gasparini proposed in "Finite Element Analysis For Minimal Shape" using a quasi-Newton method that avoids the evaluation of the Hessian matrix as required in a Newton method. They published a text and computer source code.

This text describes a mathematical model for minimizing path length, surface area and volume using a line element, a triangle element and a tetrahedron element respectively. The elements can also be used together through the unifying concept of minimizing shape volume. The square of the relative change in the element’s volume is used to define an isovolumetric element. The source and executable computer codes of the algorithm are available from the first author's website, written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD. Link: http://www.arcaro.org/main.htm

## Finite Element Analysis Software

Selected software products that enable FEM of tensegrity structures.

## Rhinoceros Software and FEM

Rhinoceros Software Can Model Tensegrity Structures Using the Rhino-Membrane Plugin.

Rhino-Membrane is a plugin for Rhinoceros 4.0, a leading 3D modelling software solution. The plug-in enables a finite element approach method, or shape finding, within the graphical user interface provided by Rhino.

A video prepared by the Rhino-Membrane team, posted at their site http://www.membranes24.com , shows how to create tensegrity structures using this form finding process. The video shows construction of a 3 strut prism, and shows previously modelled 4 strut and 5 strut prisms. The video is hosted at http://www.youtube.com/watch?v=mGaKVU8ikPc and is embedded below. The structure is modelled with elastic truss elements and predefined prestress elements.

The steps in the video:

1. Create initial geometry using polylines using tensioned cables (elastic truss elements)

2. Connect points with polylines

3. Use form finding cables. The blue vertical cables have a defined prestress instead of elastic properties

4. Convert the compressive elements to truss.

5. Set the prestrain

6. Create point objects to fix polylines

7. Fix six spatial degrees of freedom for a statically determined boundary condition

8. Define the trusses under compression (elastic truss elements) as cables

9. Assign boundary conditions

10. Assign twist angle. Every prismatic tensegrity structure has a certain "twist angle" that is independent of its height.

## Ansys Workbench

Ansys Workbench, a leading engineering simulation tool, can model tensegrity structures. Vitfc recently posted to an Ansys forum, "in order to study the efficency of tensegrity structures I am trying to write down the stiffness equations of such structures, to compare my equations with the answer of Finite Element Modelling software packages. I have done a really simply model in Ansys but I have some problems to model de behaviour of the cables in the structure. How can I model cables in Ansys Workbench? Should I choose the cables and change the properties of the material for anisotropic? or there is another way?" Below, a picture of the structure and the boundary conditions.

## References and links

Finite element method in general on wikipedia, http://en.wikipedia.org/wiki/Finite_elementArcaro's website, http://www.arcaro.org

[1] Retrieved from Snelson's Facebook Profile, 27 July 2010

[2] See Confocal-based cell-specific finite element modeling extended to study variable cell shapes and intracellular structures

Portal To MathematicsPeople: Burkhardt, Connelly, Kenner, Hart, Whiteley