PhD course

COMPUTATIONAL BIOMECHANICS: continuum models for growth and remodelling

26th-29th August, 2013

The course is lectured by professor Anders Menzel from the Institute of Mechanics, TU Dortmund, Germany and Division of Solid Mechanics, Lund University, Sweden.

Organized by the Department of Engineering Design at Tampere University of Technology and it a part of the National Doctoral Programme of Engineering Mechanics.

Download the leaflet in pdf.

Course objectives

Biological materials possess the ability to adapt according to their particular loading conditions. The course will focus on the computational continuum modeling of such phenomena, which often are denoted as growth and remodeling. As soft biological tissues – such as skin, tendons, muscles, vessels – usually undergo large deformations and possess a highly fibrous microstructure, continuum-mechanics-based modeling frameworks are commonly based on finite anisotropic elasticity. Combination of the specific constitutive models with the finite element method then allows to simulate general boundary value problems. In addition to standard continuum theories, enhanced by the concept of a multiplicative decomposition, the application of so-called computational micro-sphere models will be discussed in the course with application to growth as well as remodeling with focus on hard and soft biological tissues.


The participants are assumed to have a back-ground in finite deformation continuum mechanics. Some background in the finite element method is also desirable but not required.

Course program

  1. Introduction to one-dimension continuum thermodynamics, a primer on computational inelasticity and mass growth
  2. Growth and remodeling, three-dimensional continuum mechanics framework, multiplicative decomposition and fiber reorientation
  3. A computational micro-sphere model, application to mass growth of biological tissue and structural design
  4. A computational micro-sphere model, application to remodeling of biological tissue

Additional MATLAB tutorials

  1. Implementation of a one-dimensional growth model based on mass density evolution
  2. Implementation of a three-dimensional growth model based on a multiplicative decomposition
  3. Implementation of a three-dimensional remodeling framework based on fiber reorientation
  4. Implementation of a growth model based on a three-dimensional micro-sphere formulation

Course organization

Course Materials
The course material will consist of copies of the lecture notes and related articles. Selected related articles
E. Kuhl, A. Menzel and P. Steinmann, Computational modeling of growth, Comput. Mech., 32(1-2):71-88, 2003
A. Menzel, Modelling of anisotropic growth in biological tissues, Biomechan. Model. Mechanobiol., 3(3):147-171, 2005
A. Menzel, A fibre reorientation model for orthotropic multiplicative growth, Biomechan. Model. Mechanobiol., 6(5):303-320, 2007
A. Menzel, E. Kuhl, Frontiers in growth and remodeling, Mech. Res. Comm., 42:1-14, 2012
A. Menzel, T. Waffenschmidt, A micro-sphere-based remodelling formulation for anisotropic biological tissues, Phil. Trans. R. Soc. A., 367(1902):3499-3523, 2009
T. Waffenschmidt and A. Menzel, Application of an anisotropic growth and remodelling formulation to computational structural design, Mech. Res. Comm., 42:77-86, 2012.
T. Waffenschmidt, A. Menzel and E. Kuhl, Anisotropic density growth of bone – a computational micro-sphere approach, Int. J. Solids Struc., 49(14): 1928-1946, 2012.

Suggested Monographs
J.P. Boehler, editor. Applications of Tensor Functions in Solid Mechanics. Number 292 in CISM Courses and Lectures. Springer, 1987.
J. Bonet and R.D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge, 2008.
S.C. Cowin and S.B. Doty, Tissue Mechanics, Springer, 2007.
J.D. Humphrey, Cardiovascular Solid Mechanics, Springer, 2002.
A.J. Roberts. A one-dimensional introduction to continuum mechanics. World Scientific, 1994.
J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer, 1998.

Requirements and credits (ECTS)
Attending lectures and successful completion of home exercises will give 5 credit points. Passed examination will give additional 3 credits.

Inquiries about the course can be directed to Reijo Kouhia tel: +358(0)40 8490561, e-mail:
Registrations for the course will be taken care by the course secretary Anneli Lehtinen tel: +358 (0)40 849 0953, e-mail:

Course program

Monday: 9.15-12.00, lecture hall K1241
Tuesday: 9.15-12.00, lecture hall K1241
Wednesday: 9.15-12.00, lecture hall K1702
Thursday: 9.15-12.00, lecture hall K4108

The lectures will be given in the Department of Mechanics and Design, Korkeakoulunkatu 6, Tampere (Konetalo building is the number 6 in the map below).

Arriving to Hervanta and TUT

Buses 13 and 20 from the railway station of Tampere.
Journey planner
See also Route to TUT.

Economy accommodation

Hotel Hermica 300 meters from the university.
Omenahotels at Hämeenkatu 7 and Hämeenkatu 28 (City center, 8 km to TUT)
Hotel Ville at Hatanpään valtatie 40.
Hostel Sofia at City center.

Last update 4.6.2013