Coupled CFD/CSD Simulations of Dust Production by Fragmenting Charges Using Stabilized Linear Elements
Dr. Orlando A. Soto; Applied Simulations Inc. (ASI); McLean, Virginia, USA
Dr. Joseph D. Baum;
Applied Simulations Inc. (ASI); McLean, Virginia, USA
Prof. Rainald Löhner; George Mason University; Fairfax, Virginia, USA
Keywords: Stabilization, mixed strain/displacement formulation, CFD/CSD coupled problems, air-blast, blast propa-
gation, numerical simulations, dust effects.
Abstract
This paper presents a mixed Strain/Displacement Finite Element (FE) approach, which has been used for fracture
computation in strongly coupled blast problems. The main difference with the standard irreducible formulation (dis-
placement-based formulations) is that the tensorial strain field is one of the main FE variables of the discretized prob-
lem, hence, its rate of convergence is one order higher than the strain field obtained from standard displacement
formulations. Since the strain (or strain rate) is the main variable to compute damage and fracture of materials, a more
accurate computation of such a field gives more confident results in practical problems. Furthermore, numerical ex-
perience has shown that low order approximation of the strain field may produce totally non-physical and mesh de-
pendent fracture results. An additional benefit of the mixed strain/displacement formulation presented below, is that
it is stable for linear elements, which are very attractive due to its low computational cost, and its relatively fast pre-
processing stage (CAD and mesh generation).
Finally, the paper analyses and compares the numerical results of a coupled Computational Fluid Dynamics (CFD) /
Coupled Structural Dynamics (CSD) blast simulation with experimental data. The simulation consisted of the response
of two reinforced concrete walls to loads from a cased charge, placed in close proximity to the center of one of the
walls. This real-life case also shows the importance of taking into account the dust production due to the concrete
fragmentation, which absorbs energy from the flow, and damps in a very dramatic way the shock strength.
Introduction
Theoretical modeling and computational resolution of the strain localization process up to structural failure remains
an open challenge in computational solid dynamics (CSD). To date, most attempts to model discontinuities with stand-
ard local approaches produce non-physical solutions, which are fully determined by mesh resolution and orientation.
Cervera et al. (see [1]) showed this must be due to the poorly numerical approximation that is obtained if irreducible
formulations are used (standard displacement formulations). The previous statement may be simply explained by tak-
ing into account that in irreducible formulations, the strain, which is the variable of most interest for fracture predic-
tion, are obtained by differentiation of the fundamental unknowns (the displacement field). Hence, if linear (or tri-
linear) FE are used, the strain field has a theoretical convergence of order O(h) in L2-norm (h is the mesh size).
Therefore, the strain field has zero point convergence order (in L∞-norm), which means that even though the mesh
resolution is improved, point values do not converge. Since point strains and/or stresses (values at integration points)
are used to predict material damage and element fracture, it is of no surprise that localization bands strongly depends
on the mesh size and orientation. Contrariwise, when using the strain and displacement fields as primary variables of
the formulation, the added accuracy and convergence seems to be enough to satisfactorily solve the mentioned mesh
dependency problem (see [1] and references therein).
Herein an explicit, strain/displacement, large-deformation FE formulation to deal with strong coupled CFD/CSD
(computational fluid dynamics/computational solid dynamics) problems is presented. It is widely known that, if stand-
ard equal interpolation is used for the spatial discretization of both fields, strain and displacement, the scheme locks
and produces meaningless and non-stable results since the inf-sup condition is not fulfilled. However, equal continu-
ous FE functions are highly desirable from a computational point of view. Therefore, to circumvent the severe re-
strictions imposed by such a mathematical condition, in this work the weak forms of the mixed strain/displacement
solid dynamic equations are obtained by a variational multiscale stabilization (VMS) approach. Time discretization
of the final continuous forms is achieved by an explicit Newmark scheme, and the spatial one by using Q1/Q1 (hexa-
hedrons) or P1/P1 (tetrahedrons) standard functions. Several VMS methods were developed in [1-3] for the small-
deformation static solid equations, and successfully applied to localization problems: Totally physical and mesh inde-
pendent solutions were obtained where the standard displacement formulation failed miserably.
Finally, the CSD approach is loosely coupled with the widely tested CFD code FEFLO to solve real blast and impact
problems (see [4]). A benchmark case and one real application with dust production is presented.
