I recently joined the NL24 – Nonlinear Computational Solid & Structural Mechanics – course in Pavia. A very high scientific level course due to the expertise of the professors and to the discussed topics, ranging from classical basics to the most advanced state-of-the-art of linear and nonlinear computational mechanics.

NL24

The course aims to equip engineers, graduate students, and researchers with a comprehensive understanding of numerical methods and solution strategies for nonlinear mechanics. It will delve into the cutting-edge techniques in finite element modeling for nonlinear solid and structural mechanics, shedding light on the challenges and potential remedies across various applications.

Participants will systematically explore different sources of nonlinear behavior, focusing on aspects like nonlinear material constitutive models, large deformations, structural rotations, contact mechanics, and instabilities such as buckling. Emphasis will be placed on grasping the intricacies of structural design vulnerabilities.

Furthermore, the course will offer insights into advanced mathematical concepts and practical implementation of computational techniques. This includes finite element method, isogeometric analysis, meshless techniques, and virtual element methods.

Lessons give us a robust foundation in utilizing computational tools and software for optimizing designs and conducting detailed analyses of nonlinear structural behaviors. This also includes multiphysics and multi-scale effects, which hold significant promise for breakthroughs in various industrial sectors, from aeronautics to biomechanics.

FEAP

FEAP stands for Finite Element Analysis Program, a versatile tool tailored for both research and educational purposes. It was developed in Berkeley, university of California. Its full source code is accessible for compilation on various operating systems. The program offers a plethora of features such as defining one, two, and three-dimensional meshes, employing a diverse range of linear and nonlinear solution algorithms, and graphical capabilities for visualizing meshes and contouring solution values.

In addition, an extensive library of elements encompassing linear and nonlinear solids, thermal, frame, plate and shell, torsion, winkler foundation, acoustic, coupled problem, and multiple rigid body options with joint interactions. Constitutive models available include linear and finite elasticity, viscoelasticity with damage, and elasto-plasticity.

Moreover, FEAP can be seamlessly integrated with mesh generation programs capable of outputting nodal coordinates and element connection arrays. In such cases, users may need to develop custom functions to input the generated data.

Structural analysis FEAP patch test

FEAP * * 4-Element Patch Test
  9,4,1,2,2,4
MATErial,1
  SOLId
    PLANe STRAin
    ELAStic ISOTropic 1000.0 0.25
COORdinates
  1 0  0.0  0.0
  2 0  4.0  0.0
  3 0 10.0  0.0
  4 0  0.0  4.5
  5 0  5.5  5.5
  6 0 10.0  5.0
  7 0  0.0 10.0
  8 0  4.2 10.0
  9 0 10.0 10.0
ELEMents 1111254 2112365 3114587 4115698
! Blank termination record
! Blank termination record
                  ! Blank termination record
BOUNdary restraints
1011 4010 7010
FORCes
  3 0 2.5 0.0
  6 0 5.0 0.0
  9 0 2.5 0.0
END
  ! Blank termination record
  ! Blank termination record
Table 2.1: Data for Patch Test
BATCh
  FORM residual
  TANGent
  SOLVe
  DISPlacement ALL
  STREss ALL
END STOP
  MATErial
  COORdinates
  ELEMents
  BOUNdary restraints
  FORCes
  END
  BATCh
  END
  STOP
4-node mesh and 9-node mesh
9-node radial displacement contour

On that, we had the opportunity to know and discuss with lecturers such as Robert L. Taylor, Professor in the Graduate School, Department of Civil and Environmental Engineering, University of California, Berkeley. Peter Wriggers,  Professor of Mechanics at the Faculty of Mechanical Engineering, Leibniz Universität Hannover. Ferdinando Auricchio, Professor of Mechanics of Solids at the University of Pavia, Italy, Manfred Bischoff, Professor and head of the Institute for Structural Mechanics at the University of Stuttgart. Carlo Lovandina, Professor of Numerical Analysis at University of Milan. Alessandro Reali, Professor of Mechanics of Solids at the Department of Civil Engineering and Architecture of the University of Pavia, and “Fischer” Fellow at the Institute of Advanced Study of the Technical University of Munich. Giancarlo Sangalli, Professor of Numerical Analysis at the Mathematics Department of University of Pavia.