Computer simulations with isogeometric analysis (IGA) have multiple applications, from phase-field modeling to tumor-growth simulations. We focus on the alternating-directions solver (ADS) algorithm, in which the matrix equation representing a computational problem is decomposed into parallel tasks following the binary and balanced structure of an elimination tree. In this paper, we explore the possibility of running large-scale IGA simulations using linear computational cost alternating direction solvers on top of modern data-parallel cloud computing frameworks. To this end, we propose a new way of decomposition of the elimination tree which makes the IGA alternating-direction solver effectively a large graph problem suitable for modern cloud-computing frameworks. On this basis, we propose a new algorithm for isogeometric analysis alternating-directions solver based on the Pregel computational model, used for large-scale graph-processing in the cloud. We implement a cloud-native solver using this algorithm in the Apache Giraph framework, and show that it can be applied for solution of challenging higher-order PDEs. We evaluate the solver in terms of various scalability models and run configurations. The results indicate linear scalability of the proposed algorithm with respect to the number of elements in the mesh.

We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$Cp-1 basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$Cp-1 IGA spaces by introducing low-continuity $C^0$C0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, we propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA versus IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^2$p2 when considering rIGA discretizations rather than highly-continuous IGA spaces.

In this paper we present several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. In our approach a computational mesh is represented by a composite graph and mesh operations are expressed by the graph transformation rules. Each graph transformation system is responsible for different kind of operations. In particular, there is a graph transformation system expressing generation of an initial mesh, generating element matrices and elimination trees for interfacing with direct solver algorithm, a graph transformation system deciding which elements have to be further refined, as well as a graph transformation system responsible for execution of mesh refinements. These graph transformation systems are tested using a graph transformation tool (called GRAGRA), which provides a graphical environment for defining graphs, graph transformation rules and graph transformation systems. In this paper we illustrate the concepts by using an exemplary derivation for a three dimensional projection problem, based on a set of graph transformation rules.

In this paper we introduce formal definitions for several graph transformation systems modeling three dimensional h-adaptive Finite Element Method (3D h-FEM) algorithms with tetrahedral finite elements. We introduce a composite graph representation of the computational mesh and graph transformation rules expressing the mesh operations. In particular, there are graph transformation rules expressing the generation of the initial mesh consisting with tetrahedral finite elements, graph transformation rules expressing the construction of an elimination tree for interfacing with multi-frontal direct solver algorithm, graph transformation rules selecting sub-graph representing finite elements for further refinements, graph transformation rules responsible for execution of mesh refinements. We also discuss several benefits of using graph transformation system instead of classical FEM approach, including the benefits from the viewpoint of multi-frontal direct solvers.

This paper focuses on the application of hp hierarchic genetic strategy ( hp –HGS) for solution of a challenging problem, the inversion of 3D direct current (DC) resistivity logging measurements. The problem under consideration has been formulated as the global optimization one, for which the objective function (misfit between computed and reference data) exhibits multiple minima. In this paper, we consider the extension of the hp –HGS strategy, namely we couple the hp –HGS algorithm with a gradient based optimization method for a local search. Forward simulations are performed with a self-adaptive hp finite element method, hp –FEM. The computational cost of misfit evaluation by hp –FEM depends strongly on the assumed accuracy. This accuracy is adapted to the tree of populations generated by the hp –HGS algorithm, which makes the global phase significantly cheaper. Moreover, tree structure of demes as well as branch reduction and conditional sprouting mechanism reduces the number of expensive local searches up to the number of minima to be recognized. The common (direct and inverse) accuracy control, crucial for the hp –HGS efficiency, has been motivated by precise mathematical considerations. Numerical results demonstrate the suitability of the proposed method for the inversion of 3D DC resistivity logging measurements.

The first part of our paper presents a composite programmable graph grammar model for the self-adaptive two dimensional hp Finite Element Method algorithms (2D hp-FEM) with mixed triangular and rectangular finite elements. The two dimensional model is a starting point for the three dimensional model of self-adaptive hp-FEM presented in the second part of this paper. A computational mesh is represented by a composite graph. The operations performed over the mesh are expressed by the graph grammar rules. The three dimensional model is based on the extension of the two dimensional model with rectangular finite elements. In the second part of this paper, we conclude the presentation with numerical examples concerning the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL).

This paper presents a composite programmable graph grammar model of the three dimensional self-adaptive hp Finite Element Method (hp-FEM) algorithms. The computational mesh composed of hexahedral finite elements is represented by a composite graph. The operations performed over the mesh are expressed by composite graph grammar productions. The three dimensional model is based on the extension of the two dimensional model for rectangular finite elements. This paper is concluded with numerical examples, presenting the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL), the modern patterning process.

The aim of this work is to investigate prospective benefits of employing the idea agents for solving differential equations with hp-adaptive Finite Element Method (hp-FEM). Object-oriented model developed previously, was now translated to the language of agents along with the use of an agent-oriented parallel frontal equation solver (based on the Schur complement concept). In our paper we present the architecture and algorithms that stay behind great elasticity, wide applicability and fair responsiveness to a changing environment. We discuss interesting problems that arose, like methods to facilitate load balancing to the platform. Finally, in reference to our previous multi-scale works, the way of incorporating molecular nano-scale methods into the agent-oriented hp-FEM is discussed. To illustrate the above considerations, we use an exemplary linear elasticity problem of a non-uniform material.

Abstract The paper presents an extension of the composite programmable graph grammar (CP‐graph grammar) suitable for modeling the parallel direct solver algorithm utilized by the hp finite element method (hp‐FEM). In the proposed graph grammar model, the computational mesh is represented by a CP‐graph. The presented graph grammar models the solver algorithm by a set of graph grammar productions. The graph grammar model makes it possible to examine the concurrency of the algorithm by analyzing the interdependence between the atomic tasks, tasks and super‐tasks. The atomic tasks correspond to the graph grammar productions, representing basic undividable parts of the algorithms. The level of atomic tasks models the concurrency for the shared memory architectures. On the other hand, the tasks correspond to the groups of atomic tasks with predefined inter‐task communication channels. They constitute the grain for the decomposition of the parallel algorithm for the distributed memory architecture. Finally, the super‐tasks correspond to a group of tasks resulting from the execution of load balancing algorithm. The solver algorithm is tested on distributed memory linux cluster for up to 192 processors. Copyright © 2009 John Wiley & Sons, Ltd.

The new hp-HGS multi-deme, genetic strategy (hp-adaptive finite element method combined with hierarchic genetic strategy) for economic solving parametric inverse problems is presented in this paper. Inverse problems under consideration are formulated as the global optimization ones, where the objective is to express the discrepancy between the computed and measured energy. The efficiency of the proposed strategy results from coupling an adaptative accuracy of solving optimization problems with the accuracy of hp-FEM problem solver. The paper briefly reports the results of the asymptotic analysis that ensures the global search possibility and allows to compare the efficiency with the single population algorithm as well as with the instance of HGS without adaptation of the direct solver accuracy. A computational example shows the course of tuning the hp-FEM strategy for the simple L-shape domain benchmark.

The paper present the composite programmable (CP) graph grammar model of the selfadaptive hp Finite Element Method (hp-FEM) algorithm. The model incorporates the process of the initial mesh generation, direct solver execution, as well as mesh transformations resulting from selectedoptimal hp refinements. The computational mesh is represented by the CP-graph. The operations performed over the mesh are expressed by the graph grammar productions. The graph grammar model allows for the definition of the computational tasks for the Partitioning, Communication, Agglomeration and Mapping (PCAM) method of the parallelization of the self-adaptive hp-FEM algorithm.

The paper presents a general methodology for an efficient parallelization of the fully automatic hp-adaptive Finite Element Method (hp-FEM). The self-adaptive hp-FEM algorithm expressed in terms of the graph grammar productions is analyzed by utilizing the Partitioning Communication Agglomeration Mapping (PCAM) model. The computational tasks are defined over a graph model of the computational mesh. It is done for all parts of the algorithm: the generation of an initial mesh, direct solver (including the integration and elimination of degrees of freedom), mesh transformations (including the h and p refinements), as well as the selection of the optimal refinements. The computation and communication complexities of the resulting parallel algorithms are analyzed. The paper is concluded with the sequence of massive parallel computations. From the performed tests it implies that the code scales well up to 200 processors.