HAL Id: hal-01007829
https://hal.archives-ouvertes.fr/hal-01007829
Submitted on 26 Oct 2018HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de
Accounting for Geometric Non-Linearities in Real-Time
Simulation of Soft Tissues by Model Reduction
Techniques
Siamak Niroomandi, Icíar Alfaro, Elías Cueto, Francisco Chinesta
To cite this version:
Siamak Niroomandi, Icíar Alfaro, Elías Cueto, Francisco Chinesta. Accounting for Geometric Non-Linearities in Real-Time Simulation of Soft Tissues by Model Reduction Techniques. Computational Methods for Coupled Problems in Science and Engineering (COUPLED 2009), 2009, Ischia, Italy. �hal-01007829�
B. Schrefler, E. O˜nate and M. Papadrakakis (Eds) c
CIMNE, Barcelona, 2009
ACCOUNTING FOR GEOMETRIC NON-LINEARITIES IN
REAL-TIME SIMULATION OF SOFT TISSUES BY MODEL
REDUCTION TECHNIQUES.
SIAMAK NIROOMANDI∗, ICIAR ALFARO∗, ELIAS CUETO∗ AND
FRANCISCO CHINESTA†
∗Aragon Institute of Engineering Research. University of Zaragoza. Zaragoza, Spain.
E-mail: ecueto@unizar.es
†EADS Corporate International Chair. Ecole Centrale de Nantes. Nantes. France.
Key words: Real-time simulation, model reduction, asymptotic numerical methods, virtual surgery.
Abstract. Model reduction techniques have shown to constitute a valuable tool for real-time simulation in surgical environments and other fields. However, some limitations, imposed by real-time constraints, have not yet been overcome. One of such limitations is the severe limitation in time (established in 500 Hz of frequency for the resolution) that precludes the employ of Newton-like schemes. Thus, reduced models in real-time environments are actually linear, although the best set of basis functions, in statistical sense, is employed. In this work we present a technique able to deal with geometrically non-linear models, based on the employ of model reduction techniques, together with an Asymptotic Numerical Method (ANM) to expand the solution in the neighbourhood of the last equilibrium point. Examples of the performance of the technique over academic as well as surgical examples will be given.
1 INTRODUCTION
In a previous paper the authors introduced a new technique for the real-time simula-tion of non-linear tissue behavior based on a model reducsimula-tion technique known as Proper Orthogonal (POD) or Karhunen-Lo`eve decompositions. The technique is based upon the construction of a complete model (using Finite Element modelling or other numerical techniques, but data could also be extracted from experimental results, if available) and the extraction and storage of the most relevant information in order to construct a model with very few degrees of freedom, but that takes into account the highly non-linear re-sponse of most living tissues. It was applied to the simulation of palpation a human cornea composed of a nonlinear, hyperelastic matrix reinforced with collagen fibers [1]. But the reduced model is actually linear although the best set of basis vectors, in statistical sense,
S. Niroomandi, I. Alfaro, E. Cueto, F. Chinesta
is employed, if no updating of the tangent stiffness matrix is performed, as in a standard Newton-Raphson procedure. So the results had some error, more important the higer the strain is, in comparison with a standard FEM solution.
In this paper we apply an asymptotic numerical method (ANM) to simulate the non-linear behavior of soft tissues in a neighbourhood of a known equilibrium state. In this method complex equilibrium paths are sought in the form of asymptotic expansions, and they are determined by solving several linear problems with a single tangent stiffness matrix [2]. Then we apply model reduction technique (in this work we have used again POD) to the obtained system of equations. In this way the system can be solved in very short times. The organization of the paper is as following: the fundamentals of model re-duction techniques are explained in section 2. In section 3 first the asymptotic numerical method is introduced and then the formulation for geometrically-nonlinear problems is obtained. In section 4 it is shown how ANM formulation in reduced coordinates of POD can be obtained and in section 5 several examples are given to show the performance of the method.
2 MODEL REDUCTION TECHNIQUES
The essential idea of model reduction techniques in the context of Continuum Mechan-ics is to extract the most relevant information of a given simulation, that is performed off-line and stored for this particular application, in order to construct a good-quality Ritz-like basis to perform reduced-model (on-line) simulations with very few degrees of freedom [3][1]. These basis functions are global and, in statistical terms, of very good qual-ity. This is in sharp contrast with the finite element method, for instance, that employs general-purpose, locally supported, piecewise polynomial shape functions.
3 ASYMPTOTIC NUMERICAL METHODS
Nonlinear structural problems are generally solved using predictor-corrector methods, such as the very standard Newton-Raphson scheme. Such algorithms are successful for determining nonlinear solution branches. However, the computing time is usually large as compared to a linear solution. A family of asymptotic numerical methods (ANM) based on perturbation techniques and finite element method have been proposed by Damil and Potier-Ferry for computing perturbed bifurcations, and applied in computing the postbuckling behavior of elastic plates and shells. Next they have been extended to any nonlinear elastic solutions, plastic deformations, etc. For a complete review the interested reader can refer to [2, 4]. In contrast to predictor-corrector algorithms, the nonlinear equilibrium paths are determined by means of asymptotic expansions: the unknown U and the parameter λ are represented by power series expansions with respect to a control parameter a. By introducing the expansions into the equilibrium equation, the nonlinear problem is transformed into a sequence of linear problems in a recurrent manner and are solved by a very classical finite element method. The efficiency is due to the choice of
X Y
Z Y X
Z
Figure 1: Geometry of the finite element model for the human cornea.
a quadratic framework for the expansion process . Also because all the linear problems have the same stiffness matrix the method requires one matrix triangulation. Moreover one gets a continuous analytic representation of the branch which differs from the point by point representation of predictor-corrector algorithms.
The idea of ANM is to follow the complex equilibrium path of the nonlinear problem under an asymptotic expansion form in terms of a control parameter “a”. This expansion is developed in the neighborhood of a known solution (un; Sn
; λn) at step n and the series
is trancated at order N and λ is the loading parameter introduced in the weak form un+1(a) Sn+1(a) λn+1(a) = un(a) Sn(a) λn(a) + N X p=1 ap up Sp λp , (1)
where (up, Sp, λp) are unknowns. Above, (un+1(a), S
n+1(a), λn+1(a)) represents the
so-lution along a portion of the loading curve described continuously with respect to “a”. The introduction of Eq. (1) into the weak form of the problem leads to a series of linear problems with the same tangent operator. In addition, only reduced-order modes are employed in the simualtion, thus guaranteeing a minimum use of memory and CPU time.
4 NUMERICAL EXAMPLE: PALPATION OF THE CORNEA
In this example we applied 0.014N to nine neighbouring nodes located at the central region of the cornea, that was meshed using trilinear hexahedral elements. It consisted of 8514 nodes and 7182 elements. The mesh is shown in Figure 1 in two views. The material properties of the cornea were assumed to be E = 2MP a and ν = 0.48.
In this example nine modes were applied that provide decent approximation. The solution has been obtained using ANM-POD for p = 1, . . . , 6. In order to verify the results we have computed the solution by full FE Newton-Raphson method. The loading factor (λ) has been plotted versus the minimum displacement in Figure 4.
As it can be seen the results have good accuracy with Newton-Raphson solution. The deformed cornea obtained using ANM-POD for λ = 1 is shown in Figure 4(left) and the
S. Niroomandi, I. Alfaro, E. Cueto, F. Chinesta −0.80 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 u λ p fem p=1 p=2 p=3 p=4 p=5 p=6
Figure 2: The loading factor vs minimum displacement for the Pinched cornea.
X Y Z Uy 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -0.5 -0.55 -0.6 -0.65 -0.7 -0.75 -0.8 X Y Z Uy 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -0.5 -0.55 -0.6 -0.65 -0.7 -0.75 -0.8
Figure 3: uy contour of the pinched cornea obtained by ANM-POD (left) and FEM-Newton-Raphson
(right).
one obtained using full FE Newton-Raphson is depicted in Figure 4(right).
REFERENCES
[1] S. Niroomandi, I. Alfaro, E. Cueto, and F. Chinesta. Real-time deformable models of non-linear tissues by model reduction techniques. Computer Methods and Programs in Biomedicine, 91:223–231, 2008.
[2] B. Cochelin, N. Damil, and M. Potier-Ferry. The asymptotic-numerical method: an efficient perturbation technique for non-linear structural mechanics. Revue Europeenne des Elements Finis, 3(2):281–297, 1994.
[3] D. Ryckelynck, F. Chinesta, E. Cueto, and A. Ammar. On the a priori Model Re-duction: Overview and recent developments. Archives of Computational Methods in Engineering, 12(1):91–128, 2006.
[4] H. Zahrouni, M. Potier-Ferry, H. Elasmar, and N. Damil. Asymptotic numerical method for nonlinear constitutive laws. Revue europeenne des elements finis, 7(7):841– 869, 1998.
