Research

 

Currently there is one third-party fully funded DFG-Research Project:

NE 902/2-1 and SCHR 570/6-1 (Neff/Schröder)

„Polykonvexe Energiefunktionen im Rahmen der Anisotropen Elastizität“,

  1. 1.Förderperiode 2006-2008  Zwischenbericht ,

  2. 2.Förderperiode 2008-2010

My research interests can be summarized as follows:


Analysis:  Partial Differential Equations, Coercive Inequalities, Variational Methods, Elliptic Regularity, Applied Functionalanalysis, Evolution Equations, Polyconvexity, Convex Analysis


Numerics:  Finite-Element Methods, Discontinuous Galerkin, FETI-DP


Dimension Reduction:  Asymptotic Methods, Homogenisation, Gamma-Convergence, Theory of Shells, Plates and Membranes


Mathematical Aspects of Material Science, Continuum Mechanics:  Finite Strain Elasticity,  Finite Strain Plasticity, Gradientplasticity, Size-Effects, Microstructures


Extended Continuum Mechanics:  Cosserat, Micropolar and Micromorphic Models

I am grateful for longstanding prolific working relations with


Analysis:             Prof. K. Chelminski       (TU Warsaw)

Numerics:           Prof. A. Klawonn           (U Dui-Essen)

                            Prof. C. Wieners            (U Karlsruhe)

Engineering:       Prof. J. Schröder            (U Dui-Essen)


Home

Many problems in analysis, geometry, physics, engineering, and economics can be cast into the form of minimizing a functional I(u) among a class of admissible functions u. Important early examples of such functionals minimized in nature are: time of travel of a light ray (Fermat's principle in optics, 1662), action of a trajectory of a mechanical system (Hamilton's principle, 1834), and energy of the electrostatic field outside a charged body (Dirichlet's principle, Dirichlet, Kelvin, Gauss, 1840s). Minimizers of the latter problem solve Laplace's equation Δu=0, linking the calculus of variations to the theory of partial differential equations. Many fascinating modern day minimization problems can be viewed as far-reaching nonlinear extensions of Dirichlet's principle, in that minimizers solve nonlinear partial differential systems of equations.


In particular my interest is about existence and regularity of nonlinear PDE-Systems deriving from engineering applications and structural mechanics, as e.g., plasticity problems, extended continuum mechanics, shells and plates etc. Useful tools are newly derived coercive inequalities, like a generalized Korn‘s inequality or an inequality connecting the curl of a rotation to all partial derivatives. The precise amount of regularity of the (weak) solution is also a decisive information for subsequent FEM-implementations.


I have first applied successfully the fundamental notion of John Balls polyconvexity to materials undergoing large deformations with preferred directions (anisotropy). A specific application is e.g. biological tissue. Thus I have partly answered problem 2/18 of John Ball‘s „Some open problems in elasticity“: „Are there ways to verifying polyconvexity and quasiconvexity for a useful class of anisotropic stored energy functions?“ This topic is part of ongoing research, fully funded by the DFG.


I am also interested in 3D-to-2D reduction of nonlinear elasticity theory to membrane-, plate- and shell theories. A main mathematical tool here is Gamma convergence, introduced by De Giorgi and developed notably by Dal Maso, Braides and coworkers. It provides a powerful and rigorous mathematical framework to pass from a finer-scale (or higher-dimensional) variational principle to a coarser-scale (or lower-dimensional) effective variational principle.

  

Plasticity, either small strain or large strain, takes a prominent role in my investigations. I have developed a novel model for a geometrically exact description based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts with viscoelastic effects for which well-posedness is shown by the above mentioned methods. In this field this is one of the view results establishing the existence of classical, smooth solutions. In small strain plasticity, I have extended the Cosserat model to incorporate plastic effects and more recently I have used variational inequalities to show existence for a weak reformulation of gradient plasticity.

  

The mathematics of extended continuum models is another field of research. Here, I treat e.g. Cosserat and micromorphic models which may arise through certain homogenization schemes from small scale to large scale, think e.g. of a continuum description of a metallic foams. I have set up a new finite strain Cosserat model and I have shown existence of minimizers. A major challenge here is given by the nonlinear structure of the group of rotations SO(3). Recently, I have also re-examined the small strain Cosserat model with very weak curvature conditions, thus eliminating some troubling physical inconsistency problems of the model.

  1. P.Neff. On Korn’s first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A, 132:221–243, 2002.  location,   pdf,     TUD-Preprint Nr. 2080


P. Neff. Local existence and uniqueness for quasistatic finite plasticity with grain 

     boundary relaxation. Quart. Appl. Math., 63:88–116, 2005. QAM-PDF,  TUD-Preprint Nr. 2359


  1. A.Lew, P. Neff, D. Sulsky, and M. Ortiz. Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Applied Mathematics Research Express 3:73– 106, 2004. AMRX- Full Text (PDF),    TUD-Preprint Nr. 2300,        DOI


  1. P.Neff and I. Münch. Curl bounds Grad on SO(3). ESAIM: Control, Optimisation and Calculus of Variations, 14(1):148–159, 2008.          DOI,                    TUD-Preprint Nr. 2455


  1. P.Neff and K. Chelminski.A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via Γ-convergence. Interfaces and Free Boundaries, 9:455–492, 2007.  PDF, TUD-Preprint Nr. 2365   


  1. P.Neff and D. Knees. Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. SIAM J. Math. Anal. 40(1):21-43, 2008. DOI , TUD-Preprint Nr. 2520,        

Calculus of Variations and Nonlinear PDE-Systems

Numerical Methods

Attendant to formulating mathematically sound models I am interested in putting these models to work through Finite-Element simulations. While I am not coding myself I am treating FEM-convergence issues from a theoretical point of view e.g. in plasticity or in discontinuous Galerkin-Methods where I have shown optimal BV-convergence for linear elasticity. I am also interested in the numerical methods for shells, plates and membranes.

Selected contributions:

Application for third-party funding (October 2008) in the framework of

DFG-SPP 1420: Biomimetic Materials Research: Functionality by Hierarchical Structuring of Materials, together with M. Epple, A. Klawonn and A. Rösch


„Modelling, numerical simulation, optimization and experimental design of a hierarchically structured polymer.“