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We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings. 

Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size. 

Flexible thermoelectric (TE) devices hold great promise for energy harvesting and cooling applications, with increasing significance to serve as perpetual power sources for flexible electronics and wearable devices. Despite unique and superior TE properties widely reported in nanocrystals, transforming these nanocrystals into flexible and functional forms remains a major challenge. Herein, demonstrated is a transformative 3D conformal aerosol jet printing and rapid photonic sintering process to print and sinter solution‐processed Bi₂Te_2.7Se_0.3nanoplate inks onto virtually any flexible substrates. Within seconds of photonic sintering, the electrical conductivity of the printed film is dramatically improved from nonconductive to 2.7 × 10⁴S m⁻¹. The films demonstrate a room temperature power factor of 730 µW m⁻¹K⁻², which is among the highest values reported in flexible TE films. Additionally, the film shows negligible performance changes after 500 bending cycles. The highly scalable and low‐cost fabrication process paves the way for large‐scale manufacturing of flexible devices using a variety of high‐performing nanoparticle inks.

 

Solution‐processable semiconducting 2D nanoplates and 1D nanorods are attractive building blocks for diverse technologies, including thermoelectrics, optoelectronics, and electronics. However, transforming colloidal nanoparticles into high‐performance and flexible devices remains a challenge. For example, flexible films prepared by solution‐processed semiconducting nanocrystals are typically plagued by poor thermoelectric and electrical transport properties. Here, a highly scalable 3D conformal additive printing approach to directly convert solution‐processed 2D nanoplates and 1D nanorods into high‐performing flexible devices is reported. The flexible films printed using Sb₂Te₃nanoplates and subsequently sintered at 400 °C demonstrate exceptional thermoelectric power factor of 1.5 mW m⁻¹K⁻²over a wide temperature range (350–550 K). By synergistically combining Sb₂Te₃2D nanoplates with Te 1D nanorods, the power factor of the flexible film reaches an unprecedented maximum value of 2.2 mW m⁻¹K⁻²at 500 K, which is significantly higher than the best reported values for p‐type flexible thermoelectric films. A fully printed flexible generator device exhibits a competitive electrical power density of 7.65 mW cm⁻²with a reasonably small temperature difference of 60 K. The versatile printing method for directly transforming nanoscale building blocks into functional devices paves the way for developing not only flexible energy harvesters but also a broad range of flexible/wearable electronics and sensors.

 

Printing techniques using nanomaterials have emerged as a versatile tool for fast prototyping and potentially large‐scale manufacturing of functional devices. Surfactants play a significant role in many printing processes due to their ability to reduce interfacial tension between ink solvents and nanoparticles and thus improve ink colloidal stability. Here, a colloidal graphene quantum dot (GQD)‐based nanosurfactant is reported to stabilize various types of 2D materials in aqueous inks. In particular, a graphene ink with superior colloidal stability is demonstrated by GQD nanosurfactants via the π–π stacking interaction, leading to the printing of multiple high‐resolution patterns on various substrates using a single printing pass. It is found that nanosurfactants can significantly improve the mechanical stability of the printed graphene films compared with those of conventional molecular surfactant, as evidenced by 100 taping, 100 scratching, and 1000 bending cycles. Additionally, the printed composite film exhibits improved photoconductance using UV light with 400 nm wavelength, arising from excitation across the nanosurfactant bandgap. Taking advantage of the 3D conformal aerosol jet printing technique, a series of UV sensors of heterogeneous structures are directly printed on 2D flat and 3D spherical substrates, demonstrating the potential of manufacturing geometrically versatile devices based on nanosurfactant inks.