A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

TL;DR: A new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-planes discretization.

About: This article is published in Computers & Structures.The article was published on 2018-11-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Discretization & Dynamic problem.

Summary

1. Introduction

• Many mechanical systems and complex structures involve plate and shell parts whose main particularity is having a characteristic dimension (the one related to the thickness) much lower that the other ones (in-plane dimensions).
• When considering an implicit analysis, solution at each time step needs some iterations to enforce equilibrium.
• When dynamics applies on degenerated domains, like plates or shells, and no acceptable simplifying hypotheses are available for reducing their complexity to 2D, fully 3D solutions seem compulsory.
• Section 4 addresses time integration within the separated representation framework, and proposes an efficient hybrid explicit/implicit formulation.

2. An overview on separated representations

• Thus, when addressing a transient model involving the unknown field uðx; tÞ, its separated representation reads [20–22] uðx; tÞ XN i¼1 XiðxÞ TiðtÞ; ð1Þ where neither the time-dependent functions TiðtÞ nor the space functions XiðxÞ are ‘‘a priori” known.
• The spatial domain X is partially separable.
• The complexity of the PGD simulation scales with the twodimensional mesh used to solve the BVP’s in Xxy, regardless of the mesh used in the solution of the BVP defined in Xz for calculating ZiðzÞ.
• Domain decomposition within the separated space representation was accomplished in [25].

3.1. In-plane-out-of-plane separated representation

• As discussed in the previous section, with X having one dimension (the one related to the thickness) much smaller than the others involving the in-plane coordinates, an in-plane-out-ofplane separated representation seems again the most appealing route for addressing 3D discretizations while keeping the computational complexity the one characteristic of 2D discretizations.
• The separated representation constructor proceeds by computing a term of the sum at each iteration.
• With both Un;kxy and U n;k z unknown the resulting problem becomes non-linear.
• By introducing the trial and test functions into the weak form and then integrating in Xz because all the functions dependingon the thickness coordinate are known,weobtain a2Dweak formulation defined in Xxy whose discretization (by using a standard discretization strategy, e.g. finite elements) allows computing Un;kxy .
• Thus, the 3D computational cost is transformed into a sequence of 2D and 1D solutions, with the associated computing time savings [5].

4. Time discretization

• As previously commented explicit strategies are employed in many commercial codes.
• It is important emphasizing the main aim of the present work and the proposed methodology for performing it.
• As soon as 3D discretizations are considered, the characteristic size of the finite elements along the plate thickness becomes much smaller than the in-plane characteristic length, and then when considering explicit time integrations the time step needed for ensuring stability decreases with the throughof-thickness characteristic element length.
• In their former works [5,6] the authors proposed in the framework of elastostatics considering in-plane-out-of-plane separated representations that allowed reducing the computational complexity of solving a fully 3D problem to the one characteristic of 2D solutions.
• Thus, in this paper the authors analyze the intermediate procedure, the one in which the fine through-of-thickness representation is alleviated thanks to the use of the in-plane-out-of-plane space separated representation and its associated implicit unconditionally stable time integration.

4.1. Explicit-in-plane/implicit-out-of-plane hybrid scheme

• As just indicated, in order to circumvent the just referred stability issues, the authors propose an out-of-plane implicit discretization (unconditionally stable) while the in-plane discretization (implying coarser meshes) makes use of an explicit schema.
• It can be noticed that the derivatives involving the out-of-plane coordinate are treated using an implicit schema whereas an explicit one is retained for the in-plane derivatives.
• Thus, the hybrid schema is some place in between standard implicit and explicit techniques, taking profit of the advantages of both them.
• As usual in their previous works an alternated directions fixed point strategy is considered that by assuming Tkþ1 known calculates Pkþ1 and from the last updates Tkþ1.
• When assuming Tkþ1 known the test displacement reads u ðx; y; zÞ ¼ p uðx; yÞ tkþ1u ðzÞ p vðx; yÞ tkþ1v ðzÞ p wðx; yÞ tkþ1w ðzÞ 0 B@ 1 CA ¼ P Tkþ1; ð30Þ that introduced into the weak form (29) results in a 2D problem involving the in-plane coordinates that allows calculating Pkþ1.

5.1. Dynamics of an homogeneous plate

• In the first case study, the material occupying X is assumed homogeneous.
• The material properties are defined in Table 1, where E is the Young modulus, m the Poisson coefficient and q the material density.
• It is important to note that when considering fully explicit schemes, the stability is found being prescribed by the mesh size related to the thickness direction, however, when considering the hybrid schema the stability becomes given by the in-plane characteristic mesh size, that being much larger that the one related to the thickness, integration becomes more efficient.
• To validate the hybrid approach (only in what concerns accuracy and stability, because issues related to computing time savings were addressed in [5]), the computed solution is compared with both explicit and implicit 3D finite elements integration with a time step (in the explicit case) guaranteeing the integration stability.

5.2. Considering richer out-of-plane approximations

• In order to check the ability of the proposed technique for addressing richer out-of-plane representations, the authors consider that the domain depicted in Fig. 1 consists now in a laminated composed of 8 anisotropic plies ½0;45; 45;90 S.
• The applied force now writes again FðtÞ ¼ 0;0;A sinðxtÞð.
• The subscripts indicate respectively the proprieties along the longitudinal direction of the fibers (1), the in-plane transverse direction (2) and the out-of-plane direction (3).
• The authors compared the solution obtained using the hybrid strategy with the one obtained using implicit finite elements.
• Fig. 7 compares the time evolution of the vertical displacement at the middle of segment FG.

6. Analysis of computational performances

• In order to investigate the performances of the proposed technique the authors perform in this section different analyses.
• Before, the authors would like advertising on two facts.
• 1 m and with the material properties defined in Table 1, considering the same boundary conditions than in Section 5.1 and the same loading, the last illustrated in Fig, also known as Hz ¼ 0.
• For each mesh the authors compare the computing time employed by both the hybrid and the fully implicit PGD discretizations to solve the problem in the time interval ½0;400Dt , with the time-step Dt ¼ 10 53 s for all the simulations.
• Here not only stability issues are addressed but also the accuracy of the computed solutions.

7. Conclusion

• This paper proposes a new time discretization scheme for solving 3D dynamical problems defined in degenerated domains, that is, domains in which one of its characteristic dimensions is much smaller that the other ones, as it is the case when considering plates or shells.
• Such separated representation allows the use of extremely fine descriptions along the thickness direction.
• In this paper the authors circumvent such a drawback by using an implicit (unconditionally stable) through-the-thickness discretization whereas a standard explicit scheme is considered for treating the in-plane operators.
• The inclusion of progressive damage models combined with dynamical effects constitutes a work in progress, where the separated representations seems an appealing option to better represent damage effects along the laminate thickness, and where explicit time integrations are usually employed in industrial applications.

Figures

Table 4 Simulation parameters.

Fig. 7. Vertical displacement at the middle of segment FG.

Table 6 Meshes considered in the analysis of computational performances depicted in Fig. 10.

Fig. 8. Stress component rzzðtÞ at the central point of segment FG. Implicit and hybrid-based solutions are almost superimposed.

Table 2 Simulation parameters.

Fig. 5. Vertical displacement at the central point of segment FG.

Table 3 Mechanical properties of the 0 -ply.

Fig. 6. Loading applying in the composite laminate.

Fig. 11. Hybrid versus explicit PGD formulations.

Fig. 10. Comparing explicit, hybrid and implicit PGD formulations.

Table 8 Hybrid and implicit PGD integrations versus a standard implicit finite element formulation (using different time-steps).

Table 1 Material properties.

Fig. 1. Problem geometry.

Fig. 2. Loading.

Frequently asked questions

Q1. What have the authors contributed in "A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains" ?

In this paper the authors introduce a new efficient hybrid explicit/implicit in-plane-out-ofplane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-plane discretization. 

Q2. What are the future works mentioned in the paper "A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains" ?

In this paper the authors circumvent such a drawback by using an implicit ( unconditionally stable ) through-the-thickness discretization whereas a standard explicit scheme is considered for treating the in-plane operators. 

Q3. What is the main handicap of explicit simulations?

The main handicap of explicit simulations is that the time step must verify the stability condition, decreasing with the element size. 

Q4. What is the purpose of the last analysis?

The last analysis aims at taking advantage of the superior stability performances of the implicit formulation, that a priori can use larger time-steps that the ones of explicit and hybrid formulations that are only conditionally stables. 

Q5. What is the way to solve dynamic problems?

When dynamics applies on degenerated domains, like plates or shells, and no acceptable simplifying hypotheses are available for reducing their complexity to 2D, fully 3D solutions seem compulsory. 

Q6. What is the purpose of this paper?

This paper proposes a new time discretization scheme for solving 3D dynamical problems defined in degenerated domains, that is, domains in which one of its characteristic dimensions is much smaller that the other ones, as it is the case when considering plates or shells. 

Q7. What is the appealing route for addressing 3D discretizations?

As discussed in the previous section, with X having one dimension (the one related to the thickness) much smaller than the others involving the in-plane coordinates, an in-plane-out-ofplane separated representation seems again the most appealing route for addressing 3D discretizations while keeping the computational complexity the one characteristic of 2D discretizations. 

Q8. What is the main disadvantage of a dynamical analysis and simulation of forming processes?

In many structural analysis and simulation of forming processes dynamical aspects cannot be neglected and then elastic models are replaced by their elastodynamics counterparts. 

Q9. How does the mesh determine the limit time-step?

In fact the mesh employed for discretizing the out-of-plane dimension (thickness) determines the limit time-step ensuring stability, and consequently it could become quickly unaffordable when refining the out-of-plane discretization. 

Q10. What is the way to solve the problem of elastostatics?

In plane-out-of-plane separated representations, revisited in the next section, allows reducing the 3D solution to a sequence of 2D (in-plane) and 1D (along the thickness) problems, as proved when considering elastostatics in plate and shell domains [5,6]. 

Q11. What is the main disadvantage of explicit schemes?

On the contrary explicit schemes do not require iteration as the nodal accelerations are solved directly, and from which velocities and displacements are calculated by simple integration. 

Q12. What was the route used for deriving beam, plate and shell theories in solid mechanics?

This was the route employed for deriving beam, plate and shell theories in solid mechanics, that were extended later to many other physics, like flows in narrow gaps, thermal or electromagnetic problems in laminates, among many others. 

Q13. How many meshes are used in the analysis?

For each mesh the authors compare the computing time employed by both the hybrid and the fully implicit PGD discretizations to solve the problem in the time interval ½0;400Dt , with the time-step Dt ¼ 10 53 s for all the simulations. 

Q14. How do the authors compare the three PGD formulations?

the authors perform a comparison between the three PGD formulations (explicit, hybrid and implicit) in the time interval ½0;400Dt , with Dt ¼ 10 7 s to ensure the stability of the explicit time integration.