18 A Plausible Description of Continuum …
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them are affected by stability problems, arising from stiff differential equations and
can be managed using very small time steps, resulting in high computational costs.
In the meantime, graphic processing unit (GPU)-based solvers and adaptive
meshes were growing, and PBD methods became popular. Knowledge of traditional forces is avoided in favor of position displacements; the problem is, therefore,
transformed in a geometric constraint between configurations. The final positions of
the particles are determined by constrained matrix transformation. Many solutions
(Rivers and James 2007; Diziol et al. 2011) have been proposed to enhance the efficiency of the method that, owing to the matrix nature, can be parallelized between
the cores of GPU to reduce calculation time.
Basically, in PBD, the displacement of a particle (discretized continuum) is determined by the position of its neighbors. Therefore, the compute of new position for
a particle set can be considered as a constrained geometrical problem leading to a
transformation operator between the matrices representing the particles configuration, C t , for a discrete set of time steps t 1 , t 2 , … t n . It should be noted as most of
PBD methods hide the dynamics inside their relationship; moreover, they ask for the
knowledge of the velocity of the particles that we try to avoid. One of the differences,
between our methods and PBD, lies in the use of the velocity of the particles often
used in PBD; this, in our opinion, hides the dynamic inside so, up to now we avoid
using them.
In our approach, we try to combine the advantages of both the continuum mechanical and the position-based approaches to describe complex physical phenomena,
trying to keep the simulation easy to implement and customizable. We have written
a complete customizable and modular algorithm easily expandable to every new
feature we would like to introduce. One of the main advantages of the proposed
algorithm is the fact that it automatically takes into account large deformation elasticity, which is a topic having an increasing role in today’s research (Ladevèze 2012;
Steigmann 2009, 2010, 2013; Dell’Isola et al. 2016; Della Corte et al. 2017; Gabriele
et al. 2014). The aim of the proposed model was to develop a suitable more general
numerical tool capable of modeling the behavior of deformable bodies and to take
into account higher gradient constituent relations.
The approach tries to combine continuum mechanical material models with a
position-based method using an explicit time integration scheme to manage complex
physical effects like isotropic and anisotropic elastic behavior as well as the effects
of lateral contraction. To reply behaviors, described by constitutive equations of the
materials (e.g., Poisson’s effect), we introduce geometric constraint on the lattice
and rules ad hoc on the displacements of the points.
Since finite element method (FEM) is a reliable and well-known numerical
approach for both classical and generalized continua (see Greco and Cuomo 2013;
Ern and Guermond 2013; Greco and Cuomo 2014; Contrafatto et al. 2012; dell’Isola
et al. 2015; Cuomo and Greco 2012; Cazzani et al. 2016; Bilotta and Turco 2009 for
applications), we compared the results with the corresponding classic mechanical
continuum case, whose equations had been solved by FEM simulations with good
agreement.
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them are affected by stability problems, arising from stiff differential equations and
can be managed using very small time steps, resulting in high computational costs.
In the meantime, graphic processing unit (GPU)-based solvers and adaptive
meshes were growing, and PBD methods became popular. Knowledge of traditional forces is avoided in favor of position displacements; the problem is, therefore,
transformed in a geometric constraint between configurations. The final positions of
the particles are determined by constrained matrix transformation. Many solutions
(Rivers and James 2007; Diziol et al. 2011) have been proposed to enhance the efficiency of the method that, owing to the matrix nature, can be parallelized between
the cores of GPU to reduce calculation time.
Basically, in PBD, the displacement of a particle (discretized continuum) is determined by the position of its neighbors. Therefore, the compute of new position for
a particle set can be considered as a constrained geometrical problem leading to a
transformation operator between the matrices representing the particles configuration, C t , for a discrete set of time steps t 1 , t 2 , … t n . It should be noted as most of
PBD methods hide the dynamics inside their relationship; moreover, they ask for the
knowledge of the velocity of the particles that we try to avoid. One of the differences,
between our methods and PBD, lies in the use of the velocity of the particles often
used in PBD; this, in our opinion, hides the dynamic inside so, up to now we avoid
using them.
In our approach, we try to combine the advantages of both the continuum mechanical and the position-based approaches to describe complex physical phenomena,
trying to keep the simulation easy to implement and customizable. We have written
a complete customizable and modular algorithm easily expandable to every new
feature we would like to introduce. One of the main advantages of the proposed
algorithm is the fact that it automatically takes into account large deformation elasticity, which is a topic having an increasing role in today’s research (Ladevèze 2012;
Steigmann 2009, 2010, 2013; Dell’Isola et al. 2016; Della Corte et al. 2017; Gabriele
et al. 2014). The aim of the proposed model was to develop a suitable more general
numerical tool capable of modeling the behavior of deformable bodies and to take
into account higher gradient constituent relations.
The approach tries to combine continuum mechanical material models with a
position-based method using an explicit time integration scheme to manage complex
physical effects like isotropic and anisotropic elastic behavior as well as the effects
of lateral contraction. To reply behaviors, described by constitutive equations of the
materials (e.g., Poisson’s effect), we introduce geometric constraint on the lattice
and rules ad hoc on the displacements of the points.
Since finite element method (FEM) is a reliable and well-known numerical
approach for both classical and generalized continua (see Greco and Cuomo 2013;
Ern and Guermond 2013; Greco and Cuomo 2014; Contrafatto et al. 2012; dell’Isola
et al. 2015; Cuomo and Greco 2012; Cazzani et al. 2016; Bilotta and Turco 2009 for
applications), we compared the results with the corresponding classic mechanical
continuum case, whose equations had been solved by FEM simulations with good
agreement.