Chein-Shan Liu

Cone of non-linear dynamical system and group preserving schemes

Abstract The first step in investigating the dynamics of a continuous time system described by a set of ordinary differential equations is to integrate to obtain trajectories. In this paper, we convert the non-linear dynamical system x = f ( x ,t), x ∈ R n , into an augmented dynamical system of Lie type X = A ( X ,t) X , X ∈ M n+1 , A ∈ so (n, 1) locally. In doing so, the inherent symmetry group and the (null) cone structure of the non-linear dynamical system are brought out; then the Cayley transformation and the Pade approximants are utilized to develop group preserving schemes in the augmented space. The schemes are capable of updating the augmented state point to locate automatically on the cone at the end of each time increment. By projection we thus obtain the numerical schemes on state space x , which have the form similar to the Euler scheme but with stepsize adaptive. Furthermore, the schemes are shown to have the same asymptotic behavior as the original continuous system and do not induce spurious solutions or ghost fixed points. Some examples are used to test the performance of the schemes. Because the numerical implementations are easy and parsimonious and also have high computational efficiency and accuracy, these schemes are recommended for use in the physical calculations.

Internal symmetry in the constitutive model of perfect elastoplasticity

Abstract Internal symmetry in the constitutive model of perfect elastoplasticity is investigated here. Using homogeneous coordinates, we convert the non-linear model to a linear system X = AX . In this way the inherent symmetry in the constitutive model of perfect elastoplasticity (in the on phase) is brought out. The underlying structure is found to be the cone of Minkowski spacetime M n+1 on which the proper orthochronous Lorentz group SO 0 (n, 1) left acts. When the plasticity mechanism is shut off by the input path, the internal symmetry is switched to a translation group T(n) acting on the closed disc D n of Euclidean space E n . Based on the group properties a Cayley transformation is developed, which updates the stress points to be automatically on the yield surface at every time increment. These results (and their generalizations to more sophisticated models) are essential for computational plasticity. As an example, the results calculated using the group-preserving scheme and the exact constitutive solutions for a rectilinear path are compared.

Internal symmetry in bilinear elastoplasticity

Abstract Internal symmetry of a constitutive model of bilinear elastoplasticity (i.e. linear elasticity combined with linear kinematic hardening–softening plasticity) is investigated. First the model is analyzed and synthesized so that a two-phase two-stage linear representation of the constitutive model is obtained. The underlying structure of the representation is found to be Minkowski spacetime, in which the augmented active states admit of a Lorentz group of transformations in the on phase. The kinematic rule of the model renders the transformation group inhomogeneous, resulting in a larger group—the proper orthochronous Poincare group.

The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner

We consider a backward heat conduction problem (BHCP) in a slab, subject to noisy data at final time. The BHCP is known to be highly ill-posed. In order to stably solve the BHCP by a numerical method, we employ a new post-conditioner in the linear system obtained by the method of fundamental solutions (MFS), and then we use the conjugate gradient method (CGM) to solve the post-conditioned linear system to determine the unknown coefficients used in the expansion by the MFS. The method can retrieve the initial data rather well, with a certain degree of accuracy. Several numerical examples of the BHCP demonstrate that the present method is applicable, even for those of strongly ill-posed problems with a large value of final time and with large noise. We also demonstrate that the CGM alone is not enough to accurately recover the initial temperature.

Prandtl-Reuss elastoplasticity: On-off switch and superposition formulae

Abstract Constitutive postulates for Prandtl-Reuss elastoplasticity are selected. Based on them, sufficient and necessary conditions for plastic irreversibility are found to be the yield condition and the straining condition. This is then analyzed and it is pointed out that Prandtl-Reuss elastoplasticity is nothing but a two-phase linear system with an on-off switch, which is operated in the pace of an intrinsic measure of plastic irreversibility. Then the temporally global concept of the switch-on time t on and the switch-off time t off and their determination and bound estimation is developed. Owing to the implicit linearity, superposition formulae for the stress response are discovered. As an example, the exact constitutive solutions for circular paths based on the superposition formulae are obtained and t on and t off are determined.

Symmetry groups and the pseudo-Riemann spacetimes for mixed-hardening elastoplasticity

Abstract The constitutive postulations for mixed-hardening elastoplasticity are selected. Several homeomorphisms of irreversibility parameters are derived, among which Xa0 and Xc0 play respectively the roles of temporal components of the Minkowski and conformal spacetimes. An augmented vector Xa:=(YQat,YQa0)t is constructed, whose governing equations in the plastic phase are found to be a linear system with a suitable rescaling proper time. The underlying structure of mixed-hardening elastoplasticity is a Minkowski spacetime M n+1 on which the proper orthochronous Lorentz group SOo(n,1) left acts. Then, constructed is a Poincare group ISOo(n,1) on space X:=Xa+Xb, of which Xb reflects the kinematic hardening rule in the model. We also find that the space (Qat,q0a) is a Robertson–Walker spacetime, which is conformal to Xa through a factor Y, and conformal to Xc:=(ρQat,ρQa0)t through a factor ρ as given by ρ(q0a)=Y(q0a)/[1−2ρ0Qa0(0)+2ρ0Y(q0a)Qa0(q0a)]. In the conformal spacetime the internal symmetry is a conformal group.

Using comparison theorem to compare corotational stress rates in the model of perfect elastoplasticity

Abstract For the simple shear problem of a perfectly elastoplastic body, we convert the non-linear governing equations into a third order linear differential system, then into a second order linear differential system, and further into a Sturm–Liouville equation. Thus Sturms comparison theorem can be employed and extended to compare the simple shear responses based on different objective corotational stress rates. It is proved that the rates of Jaumann, Green–Naghdi, Sowerby–Chu, Xiao–Bruhns–Meyers, and Lee–Mallett–Wertheimer render non-oscillatory stress responses, with the Jaumann equation as a Sturm majorant for the other four equations. For an objective corotational stress rate with the general plane spin a sufficient non-oscillation criterion is found to be that the plane spin must not exceed the shear strain rate.

Lorentz group on Minkowski spacetime for construction of the two basic principles of plasticity

Abstract We show that a model of plasticity is a necessary consequence of the two basic principles: (1) causality in the truncated future cone of the Minkowski spacetime (or its generalization) of augmented states, and (2) controllability and non-generativity in a reachable, bounded space of states. To consider the symmetry switching between PSO o (n, 1) and SE(n) due to switching on and off the plasticity mechanism, the model is reconstructed as a dynamical system on a composite space, which results from a surgery on Minkowski spacetime.

A Jordan algebra and dynamic system with associator as vector field

Abstract Dynamic system defined on a vector space V which possesses one or more constraint is found to be likely described by an associator equations system x =[ y , z , u ]≔ y · zu − u · zy . The underlying algebraic structure of such dynamic system is a Jordan algebra with non-associativity, and the resulting associator of this algebra can generate a vector field, which includes one conservative force and one dissipative force. Under certain conditions on the triplet y, z and u, the system may be refreshed as a generalized Hamiltonian system with singular non-canonical metric, or a metric system with degenerate Riemannian metric. For this new formulation, some examples are explored to demonstrate its usefulness, and then the possibility to describe the non-linear dissipative phenomena of physical systems is suggested.

A NEW SHOOTING METHOD FOR QUASI-BOUNDARY REGULARIZATION OF MULTI-DIMENSIONAL BACKWARD HEAT CONDUCTION PROBLEMS

Abstract We employ a quasi‐boundary regularization to construct a two‐point boundary value problem for multi‐dimensional backward heat conduction equations. The multidimensional backward heat conduction problem (BHCP) is renowned as severely ill‐posed because the solution does not fullly depend on the data. In order to numerically tackle the multi‐dimensional BHCP, we propose a Lie‐group shooting method (LGSM) in the time direction to find the unknown initial conditions. The pivot point is based on the establishment of a one‐step Lie group element G(T) and the construction of a generalized mid‐point Lie group element G(r). Then, by imposing G(T) = G(r) we can search for the missing initial conditions through a minimum discrepancy to the real targets by the numerical ones, in terms of the weighting factor r ? (0, 1). When numerical examples are tested, we find that the LGSM is applicable to the BHCP. Even with noisy final data, the LGSM is also robust against disturbance.