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The modeling of P-waves has essential applications in seismology. This is because the detection of the P-waves is the first warning sign of the incoming earthquake. Thus, P-wave detection is an important part of an earthquake monitoring system. In this paper, we introduce a linear computational cost simulator for three-dimensional simulations of P-waves. We also generalize our formulations and derivation for elastic wave propagation problems. We use the alternating direction method with isogeometric finite elements to simulate seismic P-wave and elastic propagation problems. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along the particular coordinate axis in the sub-steps. We show that the resulting problem matrix can be represented as a multiplication of three multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. The resulting system of linear equations can be factorized in linear O (N) computational cost in every time step of the semi-implicit method. We use our method to simulate P-wave and elastic wave propagation problems. We derive the condition for the stability for seismic waves; namely, we show that the method is stable when τ < C min{ hx,hy,hz}, where C is a constant that depends on the PDE problem and also on the degree of splines used for the spatial approximation. We conclude our presentation with numerical results for seismic P-wave and elastic wave propagation problems.

In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pathogens in the air. We can represent the concentration of pathogens in the air by “contamination” propagating from the source, by the mechanisms of advection (representing air movement) and diffusion (representing the spontaneous propagation of pathogen particles in the air). The three-dimensional time-dependent advection-diffusion equation is difficult to simulate due to the high computational cost and instabilities of the numerical methods. In this paper, we present alternating directions implicit isogeometric analysis simulations of the three-dimensional advection-diffusion equations. We introduce three intermediate time steps, where in the differential operator, we separate the derivatives concerning particular spatial directions. We provide a mathematical analysis of the numerical stability of the method. We show well-posedness of each time step formulation, under the assumption of a particular time step size. We utilize the tensor products of one-dimensional B-spline basis functions over the three-dimensional cube shape domain for the spatial discretization. The alternating direction solver is implemented in C++ and parallelized using the GALOIS framework for multi-core processors. We run the simulations within 120 minutes on a laptop equipped with i7 6700 Q processor 2.6 GHz (8 cores with HT) and 16 GB of RAM.

This paper describes the application of hypergraph grammars to drive linear computationalcost solver for grids with point singularities. Such graph grammar productions are the rstmathematical formalism used to describe solver algorithm and each of them indicates thesmallest atomic task that can be executed in parallel, which is very useful in case of parallelexecution. In particular the partial order of execution of graph grammar productions can befound, and the sets of independent graph grammar productions can be localized. They canbe scheduled set by set into shared memory parallel machine. The graph grammar basedsolver has been implemented with NIVIDIA CUDA for GPU. Graph grammar productionsare accompanied by numerical results for 2D case. We show that our graph grammar basedsolver with GPU accelerator is order of magnitude faster than state of the art MUMPSsolver.

In this paper we utilize the concept of the L2 and H1 projections used toadaptively generate a continuous approximation of an input material data inthe finite element (FE) base. This approximation, along with a correspondingFE mesh, can be used as material data for FE solvers. We begin with a brieftheoretical background, followed by description of the hp-adaptive algorithmadopted here to improve gradually quality of the projections. We investigatealso a few distinct sample problems, apply the aforementioned algorithms andconclude with numerical results evaluation.

In this paper we present multi-frontal direct solver for one dimensional isogeometric finiteelement method. The solver implementation is based on graph-grammar (GG) model. TheGG model allows to express the entire solver algorithm, including generation of frontalmatrices, maerging and eliminations, as a set of basic undividable tasks called graph grammarproductions. Having the solver algorithm expressed as GG productions we can find thepartial order of execution, and create the dependency graph allowing for scheduling of tasksinto shared memory parallel machine. We focus on the implementation of the solver withNVIDIA CUDA on the graphic processing unit (GPU). The solver has been tested for linear,quadratic, cubic and higher ordere B-splines, resulting in logarithmic scalability.

In this paper we present resistivity logging measurement simulation with the usage of two types of borehole logging devices: operating with zero frequency (direct current, DC) and those operating with higher frequencies (alternate current, AC). We perform simulations of 3D resistivity measurements in deviated wells, with the sharp angle between the borehole and formation layers. We introduce hierarchical adaptive genetic strategy $hp-HGS$ interfaced with adaptive finite element method. We apply the strategy for the solution of the inverse problem, where we identify the resistivities of the formation layers based on a given measurement. We test the strategy on both direct and accurate currect cases.

In this paper, we consider the computational cost of a multi-frontal direct solver used for the factorization of matrices resulting from a discretization of isogeometric analysis with T-splines, and analysis-suitable T-splines. We start from model projection or model heat transfer problems discretized over two-dimensional meshes, either uniformly refined or refined towards a point or an edge. These grids preserve several symmetries and they are the building blocks of more complicated grids constructed during adaptive isotropic refinement procedures. A large class of computational problems construct meshes refined towards point or edge singularities. We propose an ordering that permutes the matrix in a way that the computational cost of a multi-frontal solver executed on adaptive grids is linear. We show that analysis-suitable T-splines with our ordering, besides having other well-known advantages, also significantly reduce the computational cost of factorization with the multi-frontal direct solver. Namely, the factorization with N T-splines of order p over meshes refined to a point has a linear O(Np4) cost, and the factorization with T-splines on meshes refined to an edge has O(N2pp2) cost. We compare the execution time of the multi-frontal solver with our ordering to the Approximate Minimum Degree (AMD) and Cuthill–McKee orderings available in Octave.

The paper presents the analysis of the iterative solvers utilized to solve the non-linear problemof Step-and-Flash Imprint Lithography (SFIL) a modern patterning process. The simulationsconsists in solving molecular statics problem for the polymer network, with quadratic potentials.The model distinguishes the strong interparticle interactions between particles forminga polymer network, and weak interactions between remaining particles. It also allows for largedeformations, which all together implies the non-linear model. To illustrate the convergenceof the iterative solvers, we present snapshots of the deformation of the sample being subjectto the iterative solution. We claim that the animation is an interesting way of illustratingthe convergence of the iterative solvers.

This paper describes the concurrent multi-frontal direct solver algorithm for a multi-physicsFinite Element Method (FEM). The multi-physics FEM utilizes different element sizes aswell as polynomial orders of approximation over element edges, faces, and interiors (elementnodes). The solver is based on the concept of a node, and management of unknowns isrealized at the level of nodes. The solver is tested on a challenging multi-physis problem:acoustics coupled with linear elasticity over a 3D ball shape domain.

This paper considers the communication patterns arising from the partition of geometricaldomain into sub-domains, when data is exchanged between processors assigned to adjacentsub-domains. It presents the algorithm constructing bipartite graphs covering the graphrepresentation of the partitioned domain, as well as the scheduling algorithm utilizing thecoloring of the bipartite graphs. Specifically, when the communication pattern arises from thepartition of a 2D geometric area, the planar graph representation of the domain is partitionedinto not more than two bipartite graphs and a third graph with maximum vertex valency 2,by means of the presented algorithm. In the general case, the algorithm finds h−1 or fewerbipartite graphs, where h is the maximum number of neighbors. Finally, the task of messagescheduling is reduced to a set of independent scheduling problems over the bipartite graphs.The algorithms are supported by a theoretical discussion on their correctness and efficiency.

In this paper we present the Petri net setting the optimal order of elimination for directsolver working with hp refined finite finite element meshes. The computational mesh is representedby a graph, with graph vertices corresponding to finite element nodes. The directsolver algorithm is expressed as a sequence of graph grammar productions, attributing thegraph vertices. The Petri net dictates the order of graph grammar productions, representingthe execution of the solver algorithm over a graph representation of computational mesh.The presentation is concluded with numerical experiments performed for a model L-shapedomain.