Q2. How many elements are used in the spatial discretization of the cap?
In the spatial discretization of the cap the authors use 4 elements through the thickness and a quasi-uniform discretization of each constant-radius surface into 128 elements, for a total of 512 elements.
Q3. How many elements are used in the spatial discretization of the plate?
In the spatial discretization of the plate the authors use 2 elements through the thickness and a uniform discretization in the X2±X3 plane consisting of 25 elements, for a total of 50 elements.
Q4. How is the force distribution applied to the convex face of the cap?
The force distribution f1 is applied to the convex face of the cap, acts in the negative X1-direction, is uniform in a disk centered about the X1-axis and has a resultant given by p t ÿf ; 0; 0 where f 160 kN andp t t; 06 t6 0:001 s; 0; t > 0:001 s:( 70
Q5. What are the main reasons for conserving schemes?
Conserving schemes were developed in a weak form for both compressible and incompressible hyperelastic material models, implemented using ®nite element discretizations in space and applied to three example problems.
Q6. What is the basicdi culty of the scheme?
The basicdi culty lies in the fact that any smooth solution of this system satis®es the pointwise condition det F 1 in X, and this class of deformations cannot be approximated well by standard, low-order ®nite element spaces.
Q7. Why can't the initial boundary-value problem be interpreted as a constrained or?
Due to the incompressibility condition, the initial boundary-value problem (2) for incompressible bodies can be interpreted as a constrained or differential±algebraic evolution equation [3,34] in a function space.
Q8. Why do the authors want to consider a modi®ed quasi-incompressible?
Due to special di culties associated with direct spatial discretization of (2), the authors will instead consider a modi®ed quasi-incompressible formulation along the lines of [38,40], and generalize the time-reversible, integralpreserving time discretization schemes developed in [21] to this modi®ed formulation.
Q9. How can one obtain a strong form of (10)?
by employing a standard argument involving an integration by parts, one can obtain a strong form of (10), which could then be discretized in space by an appropriate ®nite di erence method for example.
Q10. What is the cube of dimension l 0:02 m?
1. The cube is composed of a homogeneous, elastic material of Ogden type with parameters as given in (61) and density q0 1000 kg=m3.
Q11. What is the right CauchyGreen stretch associated with the deformation u?
For hyperelastic bodies, the stress ®eld S is connected to the deformation ®eld u via a local constitutive relation of the formS u 2DW C u 3 where W C 2 R is a given strain energy density function, DW C is the derivative of W with respect to its tensor argument evaluated at C, and C u is the right Cauchy±Green stretch ®eld associated with the deformation u, namely, C FTF:
Q12. What are the main results of the computational experiments presented here and in [42,43]?
the computational experiments presented here and in [20,42,43] suggest that certain implicit schemes which preserve L, J and H possess superior stability properties than other standard implicit schemes, such as the trapezoidal rule which generally preserves only L, and the mid-point rule which generally preserves only L and J.
Q13. How long does the traction f last?
In this example, the authors suppose the plate is subject to a zero displacement boundary condition along the shaded region shown in the ®gure, is initially at rest and is subject to a traction f that vanishes after a short period of time.
Q14. In what study was it proved that coupling can lead to numerical instabilities?
In [20] it was proved that such coupling can lead to numerical instabilities, while the use of interpolated strains like Cn 1=2 completely bypasses such problems.
Q15. What is the form of the second PiolaKirchho stress ®?
In the incompressible case, the total second Piola±Kirchho stress ®eld is of the formS det C 1=2kCÿ1 4 where k X; t 2 R is a material pressure ®eld that enforces the material incompressibility constraintG C det C 1=2 ÿ 1 0 in X: 5 Since DG C 12 det C 1=2Cÿ1, the authors note that the total second Piola±Kirchho stress ®eld (4) can be written inthe form S 2kDG C :
Q16. What is the e cient solution for the quasi-incompressible case?
In the quasi-incompressible case, the authors introduce a general mixed ®nite element discretization, reduce the general discretization to a two-®eld formulation and discuss an e cient solution strategy.
Q17. What is the main result of the implicit schemes for L, J and H?
it was shown that the explicit central difference scheme is the only member of the Newmark family that preserves J. The main result in [42,43] was the construction of implicit schemes for (1) that preserve L, J and H, but at the cost of introducing a nonlinear scalar equation at each time step.
Q18. What is the result of the implicit midpoint rule?
This linearization leads to a generally nonsymmetric Jacobian in the case of the conserving schemes presented here, and a symmetric Jacobian in the case of the implicit mid-point rule.
Q19. How can the authors reduce the computational costs of the conserving schemes?
it may be possible to lessen the computational costs of these conserving schemes by employing more e cient solution strategies.