Rollups have emerged as a promising approach to improving blockchains' scalability by offloading transactions execution off-chain. Existing rollup solutions either leverage complex zero-knowledge proofs or optimistically assume execution correctness unless challenged. However, these solutions suffer from high gas costs and significant withdrawal delays, hindering their adoption in decentralized applications. This paper introduces TEERollup, an efficient rollup protocol that leverages Trusted Execution Environments (TEEs) to achieve both low gas costs and short withdrawal delays. Sequencers (system participants) execute transactions within TEEs and upload signed execution results to the blockchain with confidential keys of TEEs. Unlike most TEE-assisted blockchain designs, TEERollup adopts a practical threat model where the integrity and availability of TEEs may be compromised. To address these issues, we first introduce a distributed system of sequencers with heterogeneous TEEs, ensuring system security even if a certain proportion of TEEs are compromised. Second, we propose a challenge mechanism to solve the redeemability issue caused by TEE unavailability. Furthermore, TEERollup incorporates Data Availability Providers (DAPs) to reduce on-chain storage overhead and uses a laziness penalty mechanism to regulate DAP behavior. We implement a prototype of TEERollup in Golang, using the Ethereum test network, Sepolia. Our experimental results indicate that TEERollup outperforms zero-knowledge rollups (ZK-rollups), reducing on-chain verification costs by approximately 86% and withdrawal delays to a few minutes.
Especially when modern materials like high-strength steels and aluminium are used, springback compensation algorithms can help to shorten the products development time and cost. The two algorithms, Displacement Adjustment (DA) and SpringForward (SF) that were introduced in literature have been tested extensively but some basic questions and problems remain. An analytical model for a stretch-bending process provides many possibilities to gain more insight in those problems. Furthermore, the calculation of the forming process is much faster and doesnât suffer from stability problems. In the stretch-bending model a rectangular bar is bent to a certain radius after which the load is removed and the bar recovers elastically. To model the influence of a blankholder force, the bar can also be loaded with a tension force. An elasto-plastic material model was used. The model assumes that the stress-profile is equal along the entire bar. If the DA method is used in one step, a compensation factor is required to obtain an accurate tool geometry. The optimal compensation factor can be directly calculated for the analytical model. It was shown that this factor varies heavily with increasing tension force. When the tension force is zero, or when the force is so large that the bar is deformed entirely in the plastic region, the compensation factor is close to 1.0. When an elastic band is still present in the bar, the ideal compensation factor rises from around 1 to a value of 1.5 or 2.0 depending on the material. Iterative DA was also implemented for the analytical model. With this method no knowledge about the ideal compensation factor is required, the tool shape converges to its optimal shape with each iteration. As expected the convergence depends also heavily on the tension force in the bar, in the case of pure bending (zero tension force) or fully plastic deformation (large tension force) convergence is very fast, when an elastic band is present, convergence becomes a bit slower. Although there is no straightforward mathematical or physical proof, the iterative SF method also converges for the analytical model. Interestingly, the SF method is faster than DA, and the difference is considerable in the pure bending case. The type of material also has an influence, higher strength steels require a higher compensation factor. In order to check whether the conclusions also hold for industrial forming processes, the stretch bending process was transformed to an FE model. The loading was now carried out with ârealâ tools. Opposed to the analytical model, now the DA method performs much better, especially when the tension force is raised. In that case the SF method leads to very low improvements in shape accuracy. It was shown that SF already proposes a worse tool shape in the first iteration
Energy dissipation in systems with linear viscoelastic damping is examined. It is shown that in such viscoelastically damped systems the use of additional dissipation mechanisms (such as boundary velocity feedback or thermal coupling) may not improve the rate of energy decay. The situation where the viscoelastic stress relaxation modulus decreases to its (positive) equilibrium modulus at a subexponential rate, e.g., like (1 + t)<sup>-x</sup> + E, where Îą > 0, E > 0 is examined. In this case, the nonoscillatory modes (the so-called creep modes) dominate the energy decay rate. The results are in two parts. In the first part, a linear viscoelastic wave equation with infinite memory is examined. It is shown that under appropriate conditions on the kernel and initial history, the total energy is integrable against a particular weight if the kinetic energy component of the total energy is integrable against the same weight. The proof uses energy methods in an induction argument. Precise energy decay rates have recently been obtained using boundary velocity feedback. It is shown that the same decay rates hold for history value problems with conservative boundary conditions provided that an <i>a priori</i> knowledge of the decay rate of the kinetic energy term is assumed. In the second part, a simple linear thermo-viscoelastic system, namely, a viscoelastic wave equation coupled to a heat equation, is examined. Using Laplace transform methods, an integral representation formula for <i>W(x,s</i>), the transform of the displacement <i>w(x, t)</i>, is obtained. After analyzing the location of the zeros of the appropriate characteristic equation, an asymptotic expansion for the displacement <i>w(O,t)</i> is obtained which is valid for large <i>t</i> and the specific kernel <i>g(t) = g</i>(â) + Ă´tĂ¡-1 [over]Ă (Ă¡), 0 < Ă¡ < 1. With this expansion it is shown that the coupled system tends to its equilibrium at a slower rate than that of the uncoupled system.