The problem of sampling edge-colorings of graphs with maximum degree Δ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to Δ. Vizing's theorem guarantees the existence of a (Δ+1)-edge-coloring, raising the natural question of how to efficiently sample such edge-colorings. In this paper, we take an initial step toward addressing this question. Building on the approach of Dotan, Linial, and Peled (2020), we analyze a randomized algorithm for generating random proper (Δ+1)-edge-colorings, which in particular provides an algorithmic interpretation of Vizing's theorem. The idea is to start from an arbitrary non-proper edge-coloring with the desired number of colors and at each step, recolor one edge uniformly at random provided it does not increase the number of conflicting edges (a potential function will count the number of pairs of adjacent edges of the same color). We show that the algorithm almost surely produces a proper (Δ+1)-edge-coloring and propose several conjectures regarding its efficiency and the uniformity of the sampled colorings.
Cu–Cr-based alloys are promising for thermoelectrical applications owing to their combined electrical, thermal, and mechanical performance. While their electrical properties have been widely explored, their mechanical properties have received much less attention. Herein, we investigate the tensile mechanical response of two Cu-35Cr (%wt) metal matrix composites. The first composite was fabricated by solid state sintering and the other one by vacuum arc remelting. These two processing routes result in significant differences in microstructures. The sintered composite shows equiaxed inclusions of Cr distributed in a Cu matrix with residual pores located within the Cu matrix and at the Cu/Cr interfaces. Its cast counterpart is fully dense (no porosity) and consists of Cr dendrites embedded in a Cu matrix. Monotonic and interrupted tensile tests coupled with EBSD analyses have been conducted. 2D crystal plasticity simulations have been run using the EBSD data collected in the undeformed microstructures as inputs. The model is first validated on the cast material based on the stress-strain tensile response and EBSD measurements performed after different strain increments. Then, we investigate numerically the effect of different microstructural features (initial dislocation density in the Cu matrix, matrix porosity, interfacial porosity, phase fraction, and Cr-phase morphology) on the mechanical response of the sintered composite. The intention is to discriminate the contribution of the different microstructural features on the mechanical response of Cu-Cr composites. This numerical strategy is necessary, because it would be very difficult, if not impossible, to isolate the role of different microstructural features using experiments. We show that the two main contributors to the mechanical response of the sintered composite are the initial dislocation density present in the Cu matrix, and the interfacial pores. On the contrary matrix pores have a minor effect on the mechanical response.
This paper presents a non-invasive laser-induced fluorescence (LIF) approach for temperature mapping in microfluidic chambers and then applies it to an alternating-current electrothermal (ACET) flow considered as a representative electro-microfluidic test case. Temperature mapping remains difficult when the device height is only a few hundred micrometers and when intrusive probes would disturb either the hydrodynamic field or the applied electric field. Rhodamine B dissolved in physiological saline is therefore characterized independently in absorbance and fluorescence before being used as a temperature-sensitive tracer. At a concentration as small as 1 × 10-4mol/L, the Beer–Lambert regime is retained over the optical path of the microchannel, absorbance is weakly affected by temperature in the range of interest, and the fluorescence intensity decreases linearly with a sensitivity of approximately 1.78%/K between 20°C and 45°C. The calibrated LIF measurements are complemented by micro-particle image velocimetry measurements performed in separate experimental runs in a saline-filled microchannel equipped with a pair of coplanar electrodes. In this demonstrator, the temperature gradient, not imposed externally, is generated by Joule heating associated with conduction currents, and it is coupled with the non-uniform AC electric field to produce ACET forcing. A three-dimensional electro-thermo-hydrodynamic model, based on quasi-static electrostatics, Navier–Stokes equations, and energy equation with Joule heating and temperature-dependent electrical properties included, is used to interpret and validate the measurements. Both simulations and experiments reveal a quadripolar four-clover velocity pattern above the electrode gap. Experimentally, the peak velocity reaches 350µm/s for a 20V peak-to-peak actuation at an AC frequency of 1MHz, while fluorescence thermometry reveals moderate temperature rise, below about 10K up to 25V peak-to-peak. The present methodology thus establishes a calibrated thermal diagnostic and demonstrates its ability to assess heat transfer and flow organization in electrofluidic microsystems, thus confirming its ability to prevent denaturation of biological targets, for instance.
Energy communities (ECs) are gaining strong interest to support the energy transition with increasing number of practical implementations in Europe. Encouraged by favorable regulations—such as reduced grid tariffs for internal energy exchanges— ECs enable consumers and prosumers to share locally generated electricity and reduce their energy costs. However, in practice, these exchanges rely on the public distribution network, which may lead to cost distortions if grid usage is not adequately reflected in tariff structures. This paper investigates the economic and grid impacts of ECs through a large-scale analysis of more than 500 configurations with varying sizes, installed capacities, and shares of prosumers. Each configuration is simulated under both individual and collective management schemes, with different levels of grid tariffs applied to internal exchanges. Results show that collective management systematically reduces users' electricity bills. At the same time, internal energy exchanges can grow significantly if they are not charged by any grid tariffs, leading to higher overall grid usage while reducing the revenue collected by the distribution system operator (DSO). A 10% increase in ECs self-sufficiency results in an average 8% decrease in DSO revenue, alongside a 3% increase in total grid energy usage ad 2% increase in terms of peak power. Applying standard grid tariffs to community exchanges, identical to the purchase form retailer, reverses this trend, increasing DSO revenues and reducing grid usage. These findings highlight the need for carefully designed grid tariffs that balance incentives for EC development with fair cost recovery for network operators.
Reconstructing local stress fields in heterogeneous microstructures under non-linear, history-dependent loading remains a major computational bottleneck in multi-scale simulations. We propose a coupled LSTM-GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short-Term Memory network encodes macroscopic stress–strain sequences into a compact hidden state that captures the path-dependent constitutive response, while a physics-informed Graph Neural Network reconstructs the spatially-resolved stress field at each time step. We introduce a relative weighting strategy with linear warm-up to balance the data-driven reconstruction loss and a discrete divergence-based equilibrium penalty. This resolves the scale mismatch that prevents fixed-weight formulations from converging in the elasto-plastic regime. The model is trained on 10,000 non-proportional loading paths applied to a periodic plate-with-a-hole microstructure and von Mises elasto-plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 3.0% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high-fidelity quad-element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh-agnostic. Analysis of the LSTM hidden states suggests a low-dimensional structure related to the internal state variables of the constitutive model.
