A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

This work introduces an innovative parallel, fully-distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well-known that virtual part design and qualification in additive manufacturing requires highly-accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time-to-market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high-end distributed-memory machines. It is grounded on three building blocks: (1) Hierarchical adaptive mesh refinement with octree-based meshes; (2) a parallel strategy to model the growth of the geometry; (3) state-of-the-art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder-bed technologies. After verification against a 3D benchmark, a strong-scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part-scale simulations of metal or polymer additive manufacturing, but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

In this work we exploit agglomeration based $h$-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature $h$-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element $L^2$ projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.Comment: 78 pages, 7 figure

Scalable Algorithms for Parallel Tree-based Adaptive Mesh Refinement with General Element Types

In this thesis, we develop, discuss and implement algorithms for scalable parallel tree-based adaptive mesh refinement (AMR) using space-filling curves (SFCs). We create an AMR software that works independently of the used element type, such as for example lines, triangles, tetrahedra, quadrilaterals, hexahedra, and prisms. For triangular and tetrahedral elements (simplices) with red-refinement (1:4 in 2D, 1:8 in 3D), we develop a new SFC, the tetrahedral Morton space-filling curve (TM-SFC). Its construction is similar to the Morton index for quadrilaterals/hexa- hedra, as it is also based on bitwise interleaving the coordinates of a certain vertex of the simplex, the anchor node. Additionally, we interleave with a new piece of information, the so called type. For these simplices, we develop element local algorithms such as constructing the parent, children, or face-neighbors of a simplex, and show that most of them are constant-time operations independent of the refinement level. With SFC based partitioning it is possible that the mesh elements that are parti- tioned to one process do not form a face-connected domain. We prove the following upper bounds for the number of face-connected components of segments of the TM-SFC: With a maximum refine- ment level of L, the number of face-connected components is bounded by 2(L − 1) in 2D and 2L + 1 in 3D. Additionally, we perform a numerical investigation of the distribution of lengths of SFC segments. Furthermore, we develop a new approach to partition and repartition a coarse (input) mesh among the processes. Compared to previous methods it optimizes for fine mesh load-balance and reduces the parallel communication of coarse mesh data. We discuss the coarse mesh repartitioning algorithm and demonstrate that our method repartitions a coarse mesh of 371e9 trees on 917,504 processes (405,000 trees per process) on the Juqueen supercomputer in 1.2 seconds. We develop an AMR concept that works independently of the element type; achieving this independence by strictly distinguishing between functions that oper- ate on the whole mesh (high-level) and functions that locally operate on a single element or a small set of elements (low-level). We discuss a new approach to generate and manage ghost elements that fits into our element-type independent approach. We define and describe the necessary low-level algorithms. Our main idea is the computation of tree-to-tree face-neighbors of an element via the explicit construction of the element's face as a lower dimensional element. In order to optimize the runtime of this method we enhance the algorithm with a top-down search method from Isaac, Burstedde, Wilcox, and Ghattas, and demonstrate how it speeds up the computation by factors of 10 to 20 achieving runtimes comparable to state-of-the art implementations with fixed element types. With the ghost algorithm we build a straight-forward ripple version of the 2:1 balance algorithm. This is not an optimized version but it serves as a feasibility study for our element-type independent approach. We implement all algorithms that we develop in this thesis in the new AMR library t8code. Our modular approach allows us to reuse existing software, which we demonstrate by using the library p4est for quadrilateral and hexahedral elements. In a concurrent Bachelor's thesis by David Knapp (INS, Bonn) the necessary low-level algorithms for prisms were developed. With t8code we demonstrate that we can create, adapt, (re-)partition, and balance meshes, as well as create and manage a ghost layer. In various tests we show excellent strong and weak scaling behavior of our algorithms on up to 917,504 parallel processes on the Juqueen and Mira supercomputers using up to 858e9 mesh elements. We conclude this thesis by demonstrating how an application can be coupled with the AMR routines. We implement a finite volume based advection solver using t8code and show applications with triangular, quadrilateral, tetrahedral, and hexahedral elements, as well as 2D and 3D hybrid meshes, the latter consisting of tetrahedra, hexahedra, and prisms. Overall, we develop and demonstrate a new simplicial SFC and create a fast and scalable tree-based AMR software that offers a flexibility and generality that was previously not available

High-performance direct solution of finite element problems on multi-core processors

A direct solution procedure is proposed and developed which exploits the parallelism that exists in current symmetric multiprocessing (SMP) multi-core processors. Several algorithms are proposed and developed to improve the performance of the direct solution of FE problems. A high-performance sparse direct solver is developed which allows experimentation with the newly developed and existing algorithms. The performance of the algorithms is investigated using a large set of FE problems. Furthermore, operation count estimations are developed to further assess various algorithms. An out-of-core version of the solver is developed to reduce the memory requirements for the solution. I/O is performed asynchronously without blocking the thread that makes the I/O request. Asynchronous I/O allows overlapping factorization and triangular solution computations with I/O. The performance of the developed solver is demonstrated on a large number of test problems. A problem with nearly 10 million degree of freedoms is solved on a low price desktop computer using the out-of-core version of the direct solver. Furthermore, the developed solver usually outperforms a commonly used shared memory solver.Ph.D.Committee Chair: Will, Kenneth; Committee Member: Emkin, Leroy; Committee Member: Kurc, Ozgur; Committee Member: Vuduc, Richard; Committee Member: White, Donal

A high-order finite-volume method for atmospheric flows on unstructured grids

This paper presents an extension of a Weighted Essentially Non-Oscillatory (WENO) type schemes for the compressible Euler equations on unstructured meshes for stratified atmospheric flows. The schemes could be extended for regional and global climate models dynamical cores. Their potential lies in their simplicity; accuracy; robustness; non-oscillatory properties; versatility in handling any type of grid topology; computational and parallel efficiency. Their defining characteristic is a non-linear combination of a series of high-order reconstruction polynomials arising from a series of reconstruction stencils. In the present study an explicit Strong Stability Preserving (SSP) Runge-Kutta 3rd-order method is employed for time advancement. The WENO schemes (up to 5th-order) are applied to the two dimensional and three dimensional test cases: a 2D rising thermal bubble; the 2D density current and the 3D Robert smooth bubble. The parallel performance of the schemes in terms of scalability and efficiency is also assessed