The computational burden of an explicit dynamic analysis of thin-walled structures discretized with solid-shell

elements can be very high, since the stability condition leads to extremely low time steps because of the small

thickness. A selective mass scaling procedure ([1], [2],[3]) can be introduced to overcome this limitation. The

technique proposed in [4] for single-layer 8-node solid-shell elements is here generalized to the case of multi-layer

shells. The idea is to modify the mass matrix, scaling down the highest structural eigenfrequencies, so that the

critical time step is determined only by the in-plane size of the elements, as with standard four-nodes shell meshes.

Moreover, the resulting critical time step is shown to be independent of the number of layers used for the throughthe-

thickness discretization. The accuracy of the proposed procedure and the computational gain are tested with the

aid of numerical examples

Confalonieri, Federica

Ghisi, Aldo

Perego, Umberto

The computational burden of an explicit dynamic analysis of thin-walled structures discretized with solid-shell
elements can be very high, since the stability condition leads to extremely low time steps because of the small
thickness. A selective mass scaling procedure ([1], [2],[3]) can be introduced to overcome this limitation. The
technique proposed in [4] for single-layer 8-node solid-shell elements is here generalized to the case of multi-layer
shells. The idea is to modify the mass matrix, scaling down the highest structural eigenfrequencies, so that the
critical time step is determined only by the in-plane size of the elements, as with standard four-nodes shell meshes.
Moreover, the resulting critical time step is shown to be independent of the number of layers used for the throughthe-
thickness discretization. The accuracy of the proposed procedure and the computational gain are tested with the
aid of numerical examples

CONFALONIERI, FEDERICA

GHISI, ALDO FRANCESCO

PEREGO, UMBERTO

Archivio istituzionale della ricerca - Politecnico di Milano

YIC GACM 20153rd ECCOMAS Young Investigators Conference6th GACM ColloquiumJuly 20–23, 2015, Aachen, GermanySelective mass scaling for multi-layer solid-shell discretization of thin-walledstructuresF. Confalonieri a,∗, A. Ghisi a, U. Perego aa Department of Civil and Environmental Engineering, Politecnico di MilanoPiazza Leonardo da Vinci, 32, Milano, Italy∗federica.confalonieri@polimi.itAbstract. The computational burden of an explicit dynamic analysis of thin-walled structures discretized withsolid-shell elements can be very high, since the stability condition leads to extremely low time steps because of thesmall thickness. A selective mass scaling procedure can be introduced to overcome this limitation. The techniqueproposed by Cocchetti et al. for single-layer 8-node solid-shell elements is here generalized to the case of multi-layershells.Keywords: Explicit dynamics; selective mass-scaling; solid-shell elements, multi-layer.1 INTRODUCTIONThe use of solid-shell elements is particularly suitable for simulating the mechanical response of multi-layered shellstructure, in particular when fracture and delamination phenomena are involved. The presence of different layerscan be modelled employing one or more elements per layer through the thickness. In addition, they allow for theimplementation of fully three-dimensional constitutive behaviours, being formulated in displacement degrees offreedom only. The main drawback is the high computational burden associated to the use of solid-shell elementsin an explicit dynamic simulation framework. The Courant-Friedrichs-Lewy (CFL) condition leads, indeed, toextremely low values of the stable time step, since the thickness dimension is sensibly smaller than the in-planeones. To overcome this issue, it is possible to introduce a mass scaling, modifying the solid-shell element massmatrix in order to artificially scaling down the highest structural eigenfrequencies, without significantly altering thelowest ones. Several techniques for selective mass scaling have been developed in recent years, see, for instance,[1], [2], [3]. In this paper, the selective mass scaling procedure, proposed in [4] and [5] and specifically conceivedfor solid-shell elements, is reconsidered for application to layered shells, where several solid-shell elements are usedthrough the shell thickness. The adopted mass scaling leads to a critical time step size which is determined by theelement in-plane dimensions only, independent of the layers number, with negligible accuracy loss, both in smalland large displacement problems.2 SELECTIVE MASS SCALING PROCEDUREThe reference solid shell element is the solid 8-node brick element shown in figure 1 and characterized by a thicknessdimension significantly smaller than the in-plane size. The element middle surface is coloured in grey. Let us definethe corner fibers as the segments connecting corresponding pairs of nodes belonging to the lower and upper surfacesand the corner nodes as the four nodes belonging to the middle surface of the element. As in [4] and [5], thestarting point of the proposed procedure is the definition of a transformed set of degrees of freedom, correspondingto the variables related to the corner nodes and to the corner fibers. Let us introduce the vectors of classical andtransformed nodal accelerations, ae and âe:a =[a1−4a5−8], â =[am∆a], (1)2 F. Confalonieri et al. | Young Investigators Conference 2015Figure 1: Eight-node solid shell elementFigure 2: Fiber in a multi-layered shell structurea1−4 and a5−8 represent the vectors gathering the accelerations of the nodes belonging to the lower and uppersurface of the element, while am and ∆a collect the corner nodes and the corner fibers dofs. A linear transformationcan be introduced to map the original nodal accelerations into the transformed ones.ae = Qâe. (2)with:Q =[I12 −I12I12 I12](3)where I12 is the 12×12 identity matrix.Let us consider the balance momentum equation written in variational form by the principle of virtual work:δaTeMeae = δaTe fe, (4)By introducing the transformation 3 into equation 4, it is possible to define a transformed element mass matrix M̂e.M̂e = QTMeQ =[(mup +mlow) (mup −mlow)(mup −mlow) (mup +mlow)]e(5)being mlow and mup the diagonal matrices collecting the mass coefficients of the nodes belonging to the lower andupper surfaces, respectively. The off-diagonal terms are usually small if compared with the diagonals ones and can,thus, be neglected ([5]), obtaining a lumped form of the transformed mass matrix.The dynamical response of an inertia dominated problem is mainly ruled by element translational rigid body modes,which correspond to the corner nodes dofs. The basic idea of the selective mass scaling technique relies on a localmodification of the coefficients of the lumped transformed mass matrix. Only the coefficients corresponding tothe corner fibers dofs are increaed, so that the inertia associated to the rigid body motion is left unaltered. Let usintroduce the scaled transformed mass matrix M̂eαlumped as:M̂αelumped =[(mup +mlow)00 αe(mup +mlow)]e, (6)being αe is the element mass scaling parameter. The weak form of the momentum balance equation of an undampedelement becomes:δâTe M̂αelumpedâe = δâTe f̂e. (7)The choice of the optimal scaling parameter has been discussed in details in [5]: it is computed by showing thatmass scaling is equivalent to a geometrical scaling and imposing that the geometrical scaling is such that the elementsize in the thickness direction becomes comparable to the in-plane dimension.F. Confalonieri et al. | Young Investigators Conference 2015 3When a multi-lareyed shell problem is addressed, the fiber is a multilayer segment formed by the set of corner fibersof the solid-shell elements stacked up through the shell thickness at the same in-plane position (fig. 2). Consequently,it is necessary to introduce an assembly through the thickness to properly define the fiber degrees of freedom. Letus consider the inverse of the transformation, defined in 2.âe = aeQ−1 =12aeQT (8)being,Q−1 =12QT =12[I12 I12−I12 I12](9)and introduce it into equation 7, in order to express the weak virtual balance equation in terms of nodal degrees offreedom.δaTe Q12M̂αelumped12QT︸ ︷︷ ︸Mαeae = δaTeQT 12f̂e (10)The scaled element mass matrix of a generic solid shell element Mαe is, thus, defined as:Mαe =14(QM̂αelumpedQT)=14[(1 + αe)(mlow +mup)(1 − αe)(mlow +mup)(1 − αe)(mlow +mup)(1 + αe)(mlow +mup)] (11)Let us focus on a single fiber. The mass matrix of a single layer Mαl can be defined as the assembly of the massmatrices of the elements of the support of the fiber belonging to the same layer l:Mαl =∑eMαe =[mLLl mLUlmULl mUUl](12)being:mLLl = mUUl =∑e(1 + αe)(mupe +mlowe)(13a)mLUl = mULl =∑e(1− αe)(mupe +mlowe)(13b)The overall solution can be, thus, computed simply solving a set of subsystems in the form:Mαf af = Ff (14)where the fiber mass matrixMαf is built by assembling the mass matricesMαl arising for each layer l along the fiber,namely:Mαf =Nl∑l=1Mαl (15)The resulting scaled fiber mass matrix Mf is a tridiagonal matrix, whose dimensions are directly related to thenumber of layers, which are in general in a limited number through the shell thickness. The global mass matrixMα =∑Nff=1Mαf becomes a diagonal block matrix, each block corresponding to the degrees of freedom of asingle fiber f. Even though accelerations cannot be computed explicitly, the solution of the small linear systemproviding accelerations of nodes belonging to the same fiber is inexpensive and the small additional burden is by farcompensated by the largest stable time step which can be used in the computation.The goal of the procedure is to scaling down the highest structural eigenfrequencies, so that the critical time stepis determined only by the in-plane size of the elements, as with standard four-nodes shell meshes. Moreover, theresulting critical time step is independent of the number of layers used for the through-the-thickness discretization.4 F. Confalonieri et al. | Young Investigators Conference 2015Figure 3: Sandwich beam: geometry, material layers, boundary conditions and tip displacement.3 NUMERICAL EXAMPLE: SANDWICH CANTILEVER BEAMThe bending response of a cantilever sandwich beam is addressed in this example. As shown in figure 3, the beamis completely clamped at one side and loaded at the opposite one by a surface load linearly varying in time from 0to 0.05 MPa. The beam length is equal to 600 mm, while its rectangular cross section is 60 mm wide and 20 mmdeep. The sandwich beam is composed of two external aluminum thin face sheets (E=72400 MPa, ν=0.3, ρ = 2700kg3) and of a soft core of low-density PVC foam (E= 58 MPa, ν=0.33, ρ = 60 kg3). Both aluminum layers are 0.5mm high, while the thickness of the soft core is equal to 19 mm. Each aluminum layer has been discretized by onlyone solid-shell element through-the-thickness, while five solid-shell elements of equal thickness are stacked up tomodel the soft core. Two different values of the mass scaling parameter have to be defined since different materialsand thicknesses are present: α = 400 and α = 6.93 has been considered for the aluminum and for the soft corerespectively. The analysis is run for 0.04 s. The stable time step is equal to 1.126 µs or to 0.079 µs, depending onwhether the selective mass scaling technique is applied or not. In figure 3, the vertical displacement of the loadedend of the beam computed with the proposed technique is compared both with the numerical result obtained withoutapplying the selective mass scaling procedure and with Abaqus.4 CONCLUSIONSThe proposed scaling reduces the highest eigenfrequencies, while the lower ones, associated to the rigid bodytranslations, remain unaffected. As a result, when the dynamical behavior is governed by the lowest frequencies,the structural response is well reproduced. The adopted mass scaling leads to a critical time step size which isdetermined by the element in-plane dimensions only, independent of the layers number, with negligible accuracyloss, both in small and large displacement problems. The resulting scaled mass matrix is not perfectly diagonal.However, the introduced coupling is shown to be limited to the nodes belonging to the same fiber through thethickness, so that the additional computational burden is almost negligible and by far compensated by the larger sizeof the critical time step.ACKNOWLEDGEMENTSThe financial support by Tetra Pak Packaging Solutions is kindly acknowledged.REFERENCES[1] L. Olovsson, M. Unosson, K. Simonsson. Selective mass scaling for thin walled structures modeled with tri-linear solidelements. Computational Mechanics 34:134-136. 2004.[2] A. Tkachuk, M. Bischoff. Local and global strategies for optimal selective mass scaling. Computational Mechanics53:1197-1207. 2013.[3] A. Tkachuk, M. Bischoff. Variational methods for selective mass scaling. Computational Mechanics 52:563-570. 2013.[4] G. Cocchetti, M. Pagani, U. Perego. Selective mass scaling and critical time step estimate for explicit dynamics analyseswith solid-shell elements. Computers&Structures 127:39-52. 2013.[5] G. Cocchetti, M. Pagani, U. Perego. Selective mass scaling for distorted solid-shell elements in explicit dynamics: optimalscaling factor and stable time steo estimate. International Journal of Numerical Methods in Engineering 101:700-731.2015.

Selective mass scaling for multi-layer solid-shell discretization of thin-walled structures

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Selective mass scaling for multi-layer solid-shell discretization of thin-walled structures

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