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This paper proposes two contributions to the calculation of free surface flows using the particle finite element method (PFEM). The PFEM is based on a Lagrangian approach: a set of particles defines the fluid. Then, unlike a pure Lagrangian method, all the particles are connected by a triangular mesh. The difficulty lies in locating the free surface from this mesh. It is a matter of deciding which of the elements in the mesh are part of the fluid domain, and to define a boundary - the free surface. Then, the incompressible Navier-Stokes equations are solved on the fluid domain and the particles' position is updated using the resulting velocity vector. Our first contribution is to propose an approach to adapt the mesh with theoretical guarantees of quality: the mesh generation community has acquired a lot of experience and understanding about mesh adaptation approaches with guarantees of quality on the final mesh. We use here a Delaunay refinement strategy, allowing to insert and remove nodes while gradually improving mesh quality. We show that this allows to create stable and smooth free surface geometries. Our PFEM approach models the topological evolution of one fluid. It is nevertheless necessary to apply conditions on the domain boundaries. When a boundary is a free surface, the flow on the other side is not modelled, it is represented by an external pressure. On the external free surface boundary, atmospheric pressure can be imposed. Nevertheless, there may be internal free surfaces: the fluid can fully encapsulate cavities to form bubbles. The pressure required to maintain the volume of those bubbles is a priori unknown. We propose a multi-point constraint approach to enforce global incompressibility of those empty bubbles. This approach allows to accurately model bubbly flows that involve two fluids with large density differences, while only modelling the heavier fluid.

We propose a method for computing integrable orthogonal frame fields on planar surfaces. Frames and their symmetries are implicitly represented using orthogonally decomposable (odeco) tensors. To formulate an integrability criterion, we express the frame field's Lie bracket solely in terms of the tensor representation; this is made possible by studying the sensitivity of the frame with respect to perturbations in the tensor. We construct an energy formulation that computes smooth and integrable frame fields, in both isotropic and anisotropic settings. The user can prescribe any size and orientation constraints in input, and the solver creates and places the singularities required to fit the constraints with the correct topology. The computed frame field can be integrated to a seamless parametrization that is aligned with the frame field., Comment: Submitted to the SIAM International Meshing Roundtable Workshop 2024

Accurate modeling of moving boundaries and interfaces is a difficulty present in many situations of computational mechanics. We use the eXtreme Mesh deformation approach (X-Mesh) to simulate the interaction between two immiscible flows using the finite element method, while maintaining an accurate and sharp description of the interface without remeshing. In this new approach, the mesh is locally deformed to conform to the interface at all times, which can result in degenerated elements. The surface tension between the two fluids is added by imposing the pressure jump condition at the interface, which, when combined with the X-Mesh framework, allows us to have an exactly sharp interface. If a numerical scheme fails to properly balance surface tension and pressure gradients, it leads to numerical artefacts called spurious or parasitic currents. The method presented here is well balanced and reduces such currents down to the level of machine precision.

This work presents the mathematical foundations for the generation of integrable cross-field on 2D manifolds based on user-imposed singularity configuration. In this paper, we either use singularities that appear naturally, e.g., by solving a non-linear problem, or use as an input user-defined singularity pattern, possibly with high valence singularities that typically do not appear in cross-field computations. This singularity set is under the constraint of Abel-Jacobi's conditions for valid singularity configurations. The main contribution of the paper is the development of a formulation that allows computing an integrable isotropic 2D cross-field from a given set of singularities through the resolution of only two linear PDEs. To address the issue of possible suboptimal singularities' distribution, we also present the mathematical setting for the generation of an integrable anisotropic 2D cross-field based on a user-imposed singularity pattern. The developed formulations support both an isotropic and an anisotropic block-structured quad mesh generation. Keywords: integrable 2D cross-field, valid singularity configuration, quad layout, quad meshing

In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.

The eXtreme Mesh deformation approach (X-MESH) is a new paradigm to follow sharp interfaces without remeshing and without changing the mesh topology. Even though the mesh does not change its topology, it can follow interfaces that do change their topology (nucleation, coalescence, splitting). To make this possible, the key X-MESH idea is to allow elements to reach zero measure. This permits interface relaying between nodes as well as interface annihilation and seeding in a time continuous manner. The paper targets the Stefan phase change model in which the interface (front) is at a given temperature. Several examples demonstrate the capability of the approach., Comment: 18 pages

We propose an end-to-end pipeline to robustly generate high-quality, high-order and coarse quadrilateral meshes on CAD models. This kind of mesh enables the use of high-order analysis techniques such as high-order finite element methods or isogeometric analysis. An initial unstructured mesh is generated; this mesh contains a low number of irregular vertices but these are not necessarily aligned, causing a very dense quad layout. A T-mesh is built on the mesh which allows to modify its topology by assigning new integer lengths to the T-mesh arcs. The task of simplifying the quad layout can be formulated as an Integer Linear Program which is solved efficiently using an adequate solver. Finally, a high-order quad mesh is extracted from the optimized topology. Validation on several CAD models shows that our approach is fast, robust, strictly respects the CAD features, and achieves interesting results in terms of coarseness and quality., Comment: 11 pages, 4 figures, presented at the AIAA AVIATION 2021 conference

This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.

We propose an end-to-end pipeline to robustly generate high-quality quadrilateral meshes for complex CAD models. An initial quad-dominant mesh is generated with frontal point insertion guided by a locally integrable cross field and a scalar size map adapted to the small CAD features. After triangle combination and midpoint-subdivision into an all-quadrilateral mesh, the topology of the mesh is modified to reduce the number of irregular vertices. The idea is to preserve the irregular vertices matching cross-field singularities and to eliminate the others. The topological modifications are either local and based on disk quadrangulations, or more global with the remeshing of patches of quads according to predefined patterns. Validity of the quad mesh is guaranteed by monitoring element quality during all operations and reverting the changes when necessary. Advantages of our approach include robustness, strict respect of the CAD features and support for user-prescribed size constraints. The quad mesher, which is available in Gmsh, is validated and illustrated on two datasets of CAD models., Comment: 33 pages, 10 figures, supplemental at https://www.hextreme.eu/data/quadqs2021_supplemental.pdf

A novel algorithm that produces a quad layout based on imposed set of singularities is proposed. In this paper, we either use singularities that appear naturally, e.g., by minimizing Ginzburg-Landau energy, or use as an input user-defined singularity pattern, possibly with high valence singularities that do not appear naturally in cross-field computations. The first contribution of the paper is the development of a formulation that allows computing a cross-field from a given set of singularities through the resolution of two linear PDEs. A specific mesh refinement is applied at the vicinity of singularities to accommodate the large gradients of cross directions that appear in the vicinity of singularities of high valence. The second contribution of the paper is a correction scheme that repairs limit cycles and/or non-quadrilateral patches. Finally, a high quality block-structured quad mesh is generated from the quad layout and per-partition parameterization.

Cross field generation is often used as the basis for the construction of block-structured quadrangular meshes, and the field singularities have a key impact on the structure of the resulting meshes. In this paper, we extend Ginzburg-Landau cross field generation methods with a new formulation that allows a user to impose inner singularities. The cross field is computed via the optimization of a linear objective function with localized quadratic constraints. This method consists in fixing singularities in small holes drilled in the computational domain with specific degree conditions on their boundaries, which leads to non-singular cross fields on the drilled domain. We also propose a way to calculate the Ginzburg-Landau energy of these cross fields on the perforated domain by solving a Neumann linear problem. This energy converges to the energy of the Ginzburg-Landau functional as epsilon and the radius of the holes tend to zero. To obtain insights concerning the sum of the inner singularity degrees, we give: (i) an extension of the Ginzburg-Landau energy to the piecewise smooth domain allowing to identify the positions and degrees of the boundary singularities, and (ii) an interpretation of the Poincar\'e-Hopf theorem focusing on internal singularities.

This paper presents a methodology aiming at easing considerably the generation of high-quality meshes for complex 3D domains. We show that the whole mesh generation process can be controlled with only five parameters to generate in one stroke quality meshes for arbitrary geometries. The main idea is to build a meshsize field $h(x)$ taking local features of the geometry, such as curvatures, into account. Meshsize information is then propagated from the surfaces into the volume, ensuring that the magnitude of $\vert \nabla h \vert$ is always controlled so as to obtain a smoothly graded mesh. As the meshsize field is stored in an independent octree data structure, the function h can be computed separately, and then plugged in into any mesh generator able to respect a prescribed meshsize field. The whole procedure is automatic, in the sense that minimal interaction with the user is required. Applications examples based on models taken from the very large ABC dataset, are then presented, all treated with the same generic set of parameter values, to demonstrate the efficiency and the universality of the technique., Comment: 20 pages, 18 figures

We proposed, in a recent paper (10.1002/nme.5987), a fast 3D parallel Delaunay kernel for tetrahedral mesh generation. This kernel was however incomplete in the sense that it lacked the necessary mesh improvement tools. The present paper builds on that previous work and proposes a fast parallel mesh improvement stage that delivers high-quality tetrahedral meshes compared to alternative open-source mesh generators. Our mesh improvement toolkit includes edge removal and improved Laplacian smoothing as well as a brand new operator called the Growing SPR Cavity, which can be regarded as the mother of all flips. The paper describes the workflow of the new mesh improvement schedule, as well as the details of the implementation. The result of this research is a series of open-source scalable software components, called HXT, whose overall efficiency is demonstrated on practical examples by means of a detailed comparative benchmark with two open-source mesh generators: Gmsh and TetGen.

Triangulations are an ubiquitous input for the finite element community. However, most raw triangulations obtained by imaging techniques are unsuitable as-is for finite element analysis. In this paper, we give a robust pipeline for handling those triangulations, based on the computation of a one-to-one parametrization for automatically selected patches of input triangles, which makes each patch amenable to remeshing by standard finite element meshing algorithms. Using only geometrical arguments, we prove that a discrete parametrization of a patch is one-to-one if (and only if) its image in the parameter space is such that all parametric triangles have a positive area. We then derive a non-standard linear discretization scheme based on mean value coordinates to compute such one-to-one parametrizations, and show that the scheme does not discretize a Laplacian on a structured mesh. The proposed pipeline is implemented in the open source mesh generator Gmsh, where the creation of suitable patches is based on triangulation topology and parametrization quality, combined with feature edge detection. Several examples illustrate the robustness of the resulting implementation., Comment: Images have lower quality in order to meet arxiv requirements

Three-dimensional frame fields computed on CAD models often contain singular curves that are not compatible with hexahedral meshing. In this paper, we show how CAD feature curves can induce non meshable 3-5 singular curves and we study four different approaches that aims at correcting the frame field topology. All approaches consist in modifying the frame field computation, the two first ones consisting in applying internal constraints and the two last ones consisting in modifying the boundary conditions. Approaches based on internal constraints are shown not to be very reliable because of their interactions with other singularities. On the other hand, boundary condition modifications are more promising as their impact is very localized. We eventually recommend the 3-5 singular curve boundary snapping strategy, which is simple to implement and allows to generate topologically correct frame fields., Comment: Accepted for 28th International Meshing Roundtable

3D frame fields are auxiliary for hexahedral mesh generation. There mainly exist three ways to represent 3D frames: combination of rotations, spherical harmonics and fourth order tensor. We propose here a representation carried out by the special unitary group. The article strongly relies on \cite{du1964homographies}. We first describe the rotations with quaternions, \cite[\S 13-15]{du1964homographies}. We define and show the isomorphism between unit quaternions and the special unitary group, \cite[\S 16]{du1964homographies}. The frame field space is identified as the quotient group of rotations by the octahedral group, \cite[\S 20]{du1964homographies}. The invariant forms of the vierer, tetrahedral and octahedral groups are successively built, without using homographies \cite[\S 39]{du1964homographies}. Modifying the definition of the isomorphism between unit quaternions and the special unitary group allows to use the invariant forms of the octahedral group as a unique parameterization of the orientation of 3D frames. The parameterization consists in three complex values, corresponding to a coordinate of a variety which is embedded in a three complex valued dimensional space. The underlined variety is the model surface of the octahedral group, which can be expressed with an implicit equation. We prove that from a coordinate of the surface, we may identify all the quaternions giving the corresponding 3D frames. We show that the euclidean distance between two coordinates does not correspond to the actual distance of the corresponding 3D frames. We derive the expression of three components of a coordinate in the case of frames sharing an even direction. We then derive a way to ensure that a coordinate corresponds to the special unitary group. Finally, the attempted numerical schemes to compute frame fields are given., Comment: NOT submitted. Main reference: Patrick Du Val, 1964. Homographies, quaternions and rotations

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n., Comment: Accepted for SIGGRAPH 2019

This paper presents a new way of describing cross fields based on fourth order tensors. We prove that the new formulation is forming a linear space in $\mathbb{R}^9$. The algebraic structure of the tensors and their projections on $\mbox{SO}(3)$ are presented. The relationship of the new formulation with spherical harmonics is exposed. This paper is quite theoretical. Due to pages limitation, few practical aspects related to the computations of cross fields are exposed. Nevetheless, a global smoothing algorithm is briefly presented and computation of cross fields are finally depicted.

This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders' pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders' pyramid has no hexahedral meshes with fewer than 18 interior vertices and 17 hexahedra, and introduce a valid mesh with 44 hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders' pyramid is obtained by exhaustively listing the hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid hexahedral mesh.

This paper presents a new scalable parallelization scheme to generate the 3D Delaunay triangulation of a given set of points. Our first contribution is an efficient serial implementation of the incremental Delaunay insertion algorithm. A simple dedicated data structure, an efficient sorting of the points and the optimization of the insertion algorithm have permitted to accelerate reference implementations by a factor three. Our second contribution is a multi-threaded version of the Delaunay kernel that is able to concurrently insert vertices. Moore curve coordinates are used to partition the point set, avoiding heavy synchronization overheads. Conflicts are managed by modifying the partitions with a simple rescaling of the space-filling curve. The performances of our implementation have been measured on three different processors, an Intel core-i7, an Intel Xeon Phi and an AMD EPYC, on which we have been able to compute 3 billion tetrahedra in 53 seconds. This corresponds to a generation rate of over 55 million tetrahedra per second. We finally show how this very efficient parallel Delaunay triangulation can be integrated in a Delaunay refinement mesh generator which takes as input the triangulated surface boundary of the volume to mesh.

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian.

This paper presents a method to generate high quality triangular or quadrilateral meshes that uses direction fields and a frontal point insertion strategy. Two types of direction fields are considered: asterisk fields and cross fields. With asterisk fields we generate high quality triangulations, while with cross fields we generate right-angled triangulations that are optimal for transformation to quadrilateral meshes. The input of our algorithm is an initial triangular mesh and a direction field calculated on it. The goal is to compute the vertices of the final mesh by an advancing front strategy along the direction field. We present an algorithm that enables to efficiently generate the points using solely information from the base mesh. A multi-threaded implementation of our algorithm is presented, allowing us to achieve significant speedup of the point generation. Regarding the quadrangulation process, we develop a quality criterion for right-angled triangles with respect to the local cross field and an optimization process based on it. Thus we are able to further improve the quality of the output quadrilaterals. The algorithm is demonstrated on the sphere and examples of high quality triangular and quadrilateral meshes of coastal domains are presented.

Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute B\'ezier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 B\'ezier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, \ie write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron., Comment: 13 pages, 7 figures. Submitted to the 26th International Meshing Roundtable conference. V2: removed Appendix "Derivatives of the Jacobian determinant of a linear hexahedron" and update acknowledgements. V3: modifications in abstract, introduction and conclusion

This paper proposes a method to compute crossfields based on the Ginzburg-Landau theory. The Ginzburg-Landau functional has two terms: the Dirichlet energy of the distribution and a term penalizing the mismatch between the fixed and actual norm of the distribution. Directional fields on surfaces are known to have a number of critical points, which are properly identified with the Ginzburg-Landau approach: the asymptotic behavior of Ginzburg-Landau problem provides well-distributed critical points over the 2-manifold, whose indices are as low as possible. The central idea in this paper is to exploit this theoretical background for crossfield computation on arbitrary surfaces. Such crossfields are instrumental in the generation of meshes with quadrangular elements. The relation between the topological properties of quadrangular meshes and crossfields are hence first recalled. It is then shown that a crossfield on a surface can be represented by a complex function of unit norm with a number of critical points, i.e., a nearly everywhere smooth function taking its values in the unit circle of the complex plane. As maximal smoothness of the crossfield is equivalent with minimal energy, the crossfield problem is equivalent to an optimization problem based on Ginzburg-Landau functional. A discretization scheme with Crouzeix-Raviart elements is applied and the correctness of the resulting finite element formulation is validated on the unit disk by comparison with an analytical solution. The method is also applied to the 2-sphere where, surprisingly but rightly, the computed critical points are not located at the vertices of a cube, but at those of an anticube., Comment: Promoted version of: Proceeding for the 26th International Meshing Roundtable, IMR26, 18-21 September 2017, Barcelona, Spain

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements., Comment: Preprint submitted to CAD (26th IMR special issue)

This paper presents a fast an robust mesh generation procedure that is able to generate meshes of the earth system (ocean and continent) in matters of seconds. Our algorithm takes as input a standard shape-file i.e. geospatial vector data format for geographic information system (GIS) software. The input is initially coarsened in order to automatically remove unwanted channels that are under a desired resolution. A valid non-overlapping 1D mesh is then created on the sphere using the Euclidian coordinates system $x,y,z$. A modified Delaunay kernel is then proposed that enables to generate meshes on the sphere in a straightforward manner without parametrization. One of the main difficulty in dealing with geographical data is the over-sampled nature of coastline representations. We propose here an algorithm that automatically unrefines coastline data. Small features are automatically removed while always keeping a valid (non-overlapping) geometrical representation of the domain. A Delaunay refinement procedure is subsequently applied to the domain. The refinement scheme is also multi-threaded at a fine grain level, allowing to generate about a million points per second on 8 threads. Examples of meshes of the Baltic sea as well as of the global ocean are presented., Comment: Submitted to the 25th international meshing round table

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness., Comment: Submitted to JCP

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units., Comment: Submitted to CMAME