A Parallel Direct Method for Finite Element Electromagnetic Computations Based on Domain Decomposition

High performance parallel computing and direct (factorization-based) solution methods have been the two main trends in electromagnetic computations in recent years. When time-harmonic (frequency-domain) Maxwell\u27s equation are directly discretized with the Finite Element Method (FEM) or other Partial Differential Equation (PDE) methods, the resulting linear system of equations is sparse and indefinite, thus harder to efficiently factorize serially or in parallel than alternative methods e.g. integral equation solutions, that result in dense linear systems. State-of-the-art sparse matrix direct solvers such as MUMPS and PARDISO don\u27t scale favorably, have low parallel efficiency and high memory footprint. This work introduces a new class of sparse direct solvers based on domain decomposition method, termed Direct Domain Decomposition Method (D3M), which is reliable, memory efficient, and offers very good parallel scalability for arbitrary 3D FEM problems. Unlike recent trends in approximate/low-rank solvers, this method focuses on `numerically exact\u27 solution methods as they are more reliable for complex `real-life\u27 models. The proposed method leverages physical insights at every stage of the development through a new symmetric domain decomposition method (DDM) with one set of Lagrange multipliers. Applying a special regularization scheme at the interfaces, either artificial loss or gain is introduced to each domain to eliminate non-physical internal resonances. A block-wise recursive algorithm based on Takahashi relationship is proposed for the efficient computation of discrete Dirichlet-to-Neumann (DtN) map to reduce the volumetric problem from all domains into an auxiliary surfacial problem defined on the domain interfaces only. Numerical results show up to 50% run-time saving in DtN map computation using the proposed block-wise recursive algorithm compared to alternative approaches. The auxiliary unknowns on the domain interfaces form a considerably (approximately an order of magnitude) smaller block-wise sparse matrix, which is efficiently factorized using a customized block LDL$^T$ factorization with restricted pivoting to ensure stability. The parallelization of the proposed D3M is realized based on Directed Acyclic Graph (DAG). Recent advances in parallel dense direct solvers, have shifted toward parallel implementation that rely on DAG scheduling to achieve highly efficient asynchronous parallel execution. However, adaptation of such schemes to sparse matrices is harder and often impractical. In D3M, computation of each domain\u27s discrete DtN map ``embarrassingly parallel\u27\u27, whereas the customized block LDLT is suitable for a block directed acyclic graph (B-DAG) task scheduling, similar to that used in dense matrix parallel direct solvers. In this approach, computations are represented as a sequence of small tasks that operate on domains of DDM or dense matrix blocks of the reduced matrix. These tasks can be statically scheduled for parallel execution using their DAG dependencies and weights that depend on estimates of computation and communication costs. Comparisons with state-of-the-art exact direct solvers on electrically large problems suggest up to 20% better parallel efficiency, 30% - 3X less memory and slightly faster in runtime, while maintaining the same accuracy

Effects of date palm pollen on fertility and development of reproductive system in female Balb/C mice 

Introduction: The Phoenix dactylifera or date palm pollen (DPP) is widely used for male infertilityin traditional medicine. The aim of this study was to evaluate the effect of DPP on fertility and development of female reproductive system in Balb/C mice.Methods: Ten groups were assigned to 2 control and 8 experimental groups. On the control groups1 and 2 no treatment was conducted, but in the control group 2, after 10 days the mice were mated.In experimental groups 1 and 2, the animals received DPP (100 and 200 mg/kg, respectively) byoral administration for 10 days. In experimental groups 3 and 4 percentage of mating was evaluatedafter 10 days. Experimental groups 5 and 6 received DPP during gestation. Embryos were removedto evaluate ovaries histology. For experimental groups 7 and 8, DPP was administered until 21thday after birth. The offspring ovaries were removed to evaluate histological parameters. The levelsof sexual hormones were also measured.Results: Several parameters of ovaries in offspring, including mass index, diameter of ovaries,number of primary and secondary graph follicles and corpora luteal, percentage of mating, bodymass index and Crown rump (CR) of embryos, diameter of ovary, basic sexual cell number inembryos, and mass index increased in experimental groups in comparison to the controls.However, luteinizing hormone (LH) and follicle-stimulating hormone (FSH) in the experimentalgroups were not significantly different from those in control groups (P > 0.05), while estrogen andprogesterone considerably increased in experimental groups (P < 0.05).Conclusion: Results of our investigation propose that DPP can create an appropriate situationfor oogenesis and maintain efficient fertility in female mice which it may be considered as usefulnutraceuticals for potentiation of fertility in future human studies.&nbsp

Effects of date palm pollen on fertility and development of reproductive system in female Balb/C mice

Introduction: The Phoenix dactylifera or date palm pollen (DPP) is widely used for male infertilityin traditional medicine. The aim of this study was to evaluate the effect of DPP on fertility and development of female reproductive system in Balb/C mice.Methods: Ten groups were assigned to 2 control and 8 experimental groups. On the control groups1 and 2 no treatment was conducted, but in the control group 2, after 10 days the mice were mated.In experimental groups 1 and 2, the animals received DPP (100 and 200 mg/kg, respectively) byoral administration for 10 days. In experimental groups 3 and 4 percentage of mating was evaluatedafter 10 days. Experimental groups 5 and 6 received DPP during gestation. Embryos were removedto evaluate ovaries histology. For experimental groups 7 and 8, DPP was administered until 21thday after birth. The offspring ovaries were removed to evaluate histological parameters. The levelsof sexual hormones were also measured.Results: Several parameters of ovaries in offspring, including mass index, diameter of ovaries,number of primary and secondary graph follicles and corpora luteal, percentage of mating, bodymass index and Crown rump (CR) of embryos, diameter of ovary, basic sexual cell number inembryos, and mass index increased in experimental groups in comparison to the controls.However, luteinizing hormone (LH) and follicle-stimulating hormone (FSH) in the experimentalgroups were not significantly different from those in control groups (P > 0.05), while estrogen andprogesterone considerably increased in experimental groups (P < 0.05).Conclusion: Results of our investigation propose that DPP can create an appropriate situationfor oogenesis and maintain efficient fertility in female mice which it may be considered as usefulnutraceuticals for potentiation of fertility in future human studies