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"Formulasi skema numerik sebagai diskretisasi ruang terutama untuk persamaan diferensial parsial hiperbolik nonlinear dikembangkan secara berkelanjutan. Hal ini dilakukan untuk memperbaiki berbagai aspek yang menjadi masalah utama dalam penyelesaian persamaan tersebut, diantaranya: diskontinuitas, spektrum skala aliran yang luas, dan kestabilan numerik. Pada bagian awal penelitian ini, dilakukan formulasi alternatif untuk indikator smoothness skema targeted ENO (TENO). Studi kestabilan numerik dan approximate dispersion relation (ADR) memberikan nilai parameter q1=2, q2=3 dan q1=1, q2=6. Berdasarkan ADR ditentukan bahwa indikator smoothness skema TENO dengan parameter q1=1, q2=6 memberikan disipasi numerik paling rendah dan dispersi yang baik. Pada bagian berikutnya, diusulkan reformulasi skema TENO menggunakan polinomial Hermite. Skema Hermite TENO (HTENO) ini mempunyai beberapa keuntungan dibandingkan skema yang telah ada sebelumnya, antara lain rekonstruksi yang lebih ringkas dan mempunyai disipasi numerik yang rendah dengan dimanfaatkannya strategi pemilihan stensil berbasis skema TENO. Selanjutnya, diformulasikan global reference smoothness indicator baru untuk skema yang diusulkan. Perhitungan fluks dan integrasi waktu secara berturut-turut diselesaikan dengan metode Lax-Friedrichs lokal dan strong-stability-preserving Runge-Kutta orde ketiga. Pada penelitian ini, didemonstrasikan uji numerik skema HTENO pada persamaan skalar dan persamaan Euler untuk fluida kompresibel dalam satu- dan dua-dimensi. Pada uji numerik tersebut, skema numerik yang diusulkan memberikan disipasi numerik yang rendah, peningkatan performa penangkapan diskontinuitas dan memberikan resolusi fluktuasi aliran berskala kecil yang baik sebagaimana skema TENO.

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Formulation of the numerical schemes as spatial discretization, particularly for nonlinear hyperbolic partial differential equations, is developed continuously. Those works improve various aspects that become the main problem to solve these equations, including discontinuity, a broad spectrum of flow scales, and numerical stability. At the beginning of this research, an alternative formulation for the smoothness indicator of the targeted ENO (TENO) scheme was carried out. The study of numerical stability and approximate dispersion relation (ADR) give parameter values ​​ q1=2, q2=3 and q1=1, q2=6. Based on the ADR, TENO’s smoothness indicator with the parameters q1=1, q2=6 gives the lowest numerical dissipation and good dispersion.

In the next section, we propose a reformulation of the TENO scheme using Hermite polynomials. The Hermite TENO (HTENO) reconstructions offer major advantages over earlier reconstructions; namely, it is a compact Hermite-type reconstruction and has low dissipation by virtue of TENO’s stencil voting strategy. Next, new high-order global reference smoothness indicators for the proposed scheme are formulated. The flux calculation and time integration were carried out by using the local Lax-Friedrichs and the third-order strong-stability-preserving Runge-Kutta method, respectively. In this research, numerical tests of the HTENO scheme on scalar equations and Euler equations of compressible flow in one- and two-dimensions are demonstrated. In these tests, the proposed scheme gives low numerical dissipation, improves the shock-capturing performance and inherits the good small-scale resolution of the TENO scheme."

"Persamaan Euler merupakan salah satu penyederhanaan persamaan Navier-Stokes dengan asumsi inviscid, adiabatik serta menghilangkan efek dari body force. Pada aliran kompresibel, persamaan Euler merupakan sistem pesamaan hiperbolik non-linear untuk hukum konservasi. Pada aliran kompresibel, munculnya fenomena diskontinuitas berupa gelombang kejut sering menimbulkan masalah dalam simulasi, terutama dalam hal akurasi. Pada skema Godunov, akurasi interpolasi untuk memperoleh fluks pada batas antar sel dapat ditingkatkan dengan penggunaan limiter. Salah satu limiter orde tinggi yang dapat digunakan dalam penyelesaian persamaan Euler adalah skema weighted essentially non-oscillatory (WENO). Masalah yang timbul dari penggunaan skema WENO sebagai limiter adalah beban komputasi yang sangat tinggi, terlebih jika sistem persamaan dan domain komputasi yang kompleks. Pengurangan beban komputasi dapat dilakukan dengan cara simplifikasi skema WENO itu sendiri atau dengan menggunakan skema hibrid dimana skema WENO akan digunakan pada kondisi tertentu. Pada penelitian ini dikembangkan skema hibrid orde tinggi yang mengadopsi WENO pada daerah diskontinu dengan deteksi diskontinuitas secara lokal. Metode cell-centered finite volume digunakan untuk diskretisasi ruang. Penyelesaian masalah Riemann pada batas sel digunakan skema Harten-Lax-van Leer contact (HLLC) dan Lax-Friedrichs, serta untuk integrasi waktu digunakan skema strong stability preserving Runge-Kutta orde ketiga untuk memberikan kestabilan yang baik pada skema numerik. Berdasarkan hasil yang diperoleh, skema hibrid yang dikembangkan cukup efektif digunakan dalam penyelesaian masalah aliran kompresibel. Pengurangan waktu komputasi yang signifikan dan akurasi yang baik menjadikan skema hibrid yang dikembangkan menjadi salah satu pilihan skema numerik orde tinggi yang baik untuk dapat diterapkan dalam simulasi aliran kompresibel.

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Euler equation is a simplification of Navier-Stokes equation which assume the flows are inviscid, adiabatic, and eliminating the effects of body forces. In the compressible flow, the Euler equation is a non-linear hyperbolic conservation laws. The presence of the discontinuities phenomenon in the form of shock wave in the compressible flow often arise the problem in the simulation, mainly in the terms of accuracy. In the Godunovs scheme, the accuracy of interpolation to obtain flux at the intercell boundary can be improved by using a high order limiter. One of the high order limiter that can be used to solve the Euler equation is weighted essentially non-oscillatory (WENO) scheme.

The problem that arises from the use of WENO scheme is high computational loads, moreover the system of equations or the domain are very complex. To reduce the computational cost, it can be done by simplify the WENO reconstruction or implement the hybrid scheme where the WENO scheme only applied in certain conditions.

In this study, hybrid high order scheme are developed which adopt the WENO schem in the discontinuous region by detecting the local discontinuities. The cell-centered finite volume are used in the spatial discretization. Harten-Lax-van Leer contact (HLLC) and Lax-Friedrichs scheme are used to solve Riemann problem in the cell boundary, and third order strong stability preserving Runge-Kutta (SSP-RK) scheme is used for time integration to ensure the positivity and provide good stability in the numerical scheme.

The results shows that the hybrid scheme developed in this work are effective for solving compressible flow problem. The significant reduction of the computational cost and the satisfactory accuracy results are make this hibrid scheme become one of the good choices of high order numerical scheme to be applied in the compressible flow simulation."

"Extended finite element method provides an introduction to the extended finite element method (XFEM), a novel computational method which has been proposed to solve complex crack propagation problems. The book helps readers understand the method and make effective use of the XFEM code and software plugins now available to model and simulate these complex problems.The book explores the governing equation behind XFEM, including level set method and enrichment shape function. The authors outline a new XFEM algorithm based on the continuum-based shell and consider numerous practical problems, including planar discontinuities, arbitrary crack propagation in shells and dynamic response in 3D composite materials."

"This book is written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a tutorial for readers who are new to the topic. Following the tutorial, chapters in Part I address fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics."

"The Finite Element Method for Elliptic Problems is the only book available that analyzes in depth the mathematical foundations of the finite element method. It is a valuable reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures, and there are many exercises of varying difficulty.Although nearly 25 years have passed since this book was first published, the majority of its content remains up-to-date. Chapters 1 through 6, which cover the basic error estimates for elliptic problems, are still the best available sources for material on this topic. The material covered in Chapters 7 and 8, however, has undergone considerable progress in terms of new applications of the finite element method; therefore, the author provides, in the Preface to the Classics Edition, a bibliography of recent texts that complement the classic material in these chapters."

"ABSTRAKTeori UI-Beam adalah suatu teori baru dalam Metode Elemen Hingga (MEH) yang merupakan modifikasi dari teori Balok Timoshenko. Sementara itu Isogeometrik sendiri adalah pengembangan dari MEH yang menggunakan fungsi dari B-Splines, dimana fungsi ini menggantikan shape function dalam MEH. Kelebihan dari Isogeometrik adalah dapat menghasilkan kelengkungan geometri yang sempurna walaupun hanya menggunakan sedikit elemen. Skripsi ini berfokus pada kasus balok di atas pondasi elastis variabel tunggal, dimana hasil antara MEH dan Isogeometrik akan dibandingkan dengan eksak.

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ABSTRACT

The theory of UI-Beam happens to be the new thesis within the Finite Element Method (FEM) which is a modification of the Timoshenko Beam. Meanwhile, Isogeometric Method is a development of FEM itself that it uses B-Spline function to replace the shape function in FEM. The capability of Isogeometric Method to produce a perfect geometry curvature albeit using only a few elements becomes its adequate advantage. This paper concerns about single variable beam on the elastic foundation case which, later on, will lay a comparison between FEM, Isogeometric Method, the exact solution forward."

"ABSTRAKGWPhreatic adalah program komputer dengan algoritma metode beda hingga untuk analisis aliran air tanah yang dikembangkan oleh Departemen Teknik Sipil Universitas Indonesia untuk keperluan akademis. Program tersebut telah melalui validasi menggunakan model fisik yang dibuat oleh Purnaman 1998 , Zein 1998 , Triono 1999 , Andrias 2001 , dan Handoyo 2001 . Model fisik yang digunakan meliputi model fisik aliran air tanah dengan di antara dua badan air, aliran air tanah dengan batas kedap air di kanan dan kirinya, aliran air tanah dengan turap, dan aliran air tanah dengan imbuhan sumur dengan susunan akuifer yang beragam. Semua validasi tersebut menyatakan bahwa GWPhreatic valid untuk simulasi berbagai kasus yang digunakan. Selain validasi-validasi tersebut, ada pula pengembangan protokol fisik saja yang belum digunakan untuk validasi GWPhreatic, yaitu protokol aliran air tanah melalui akuifer berlapis oleh Kaeni 2015 dan protokol aliran air tanah melalui akuifer melayang oleh Saragih 2015 . Oleh karena itu, penulis melakukan penelitian dengan menggunakan kedua model fisik tersebut untuk mengevaluasi tingkat akurasi GWPhreatic dan membandingkannya dengan tingkat akuirasi program komersial yang sudah terpercaya. Program komersial yang digunakan sebagai pembanding adalah GeoStudio 2016 yang menggunakan algoritma metode elemen hingga sebab GeoStudio 2016 menyediakan lisensi gratis untuk keperluan akademis, meskipun dengan beberapa keterbatasan. Penelitian tersebut bertujuan untuk mengetahui seberapa besar bias antara program GWPhreatic dan GeoStudio 2016 terhadap model fisik serta mengetahui seberapa akurat GWPhreatic dibandingkan GeoStudio 2016 dalam melakukan simulasi model fisik serupa. Tingkat akurasi keduanya dinilai berdasarkan bias keduanya terhadap model fisik. Hasilnya, bias GWPhreatic pada simulasi akuifer berlapis sebesar 0,0049 dan 0,0005 pada simulasi akuifer melayang. Sedangkan bias GeoStudio pada simulasi akuifer berlapis sebesar 0,0069 dan 0,0042 pada simulasi akuifer melayang. Nilai-nilai tersebut tergolong sangat kecil, sehingga dapat disimpulkan bahwa GWPhreatic dapat diandalkan dalam melakukan simulasi aliran air tanah melalui akuifer berlapis dan akuifer melayang sebagaimana halnya GeoStudio.

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ABSTRACT

GWPhreatic is a computer program with finite difference method algorithm for phreatic groundwater flow analysis that is developed by Department of Civil Engineering of Universitas Indonesia for academic purposes. This program has been going through several validations with physical models made by Purnaman 1998 , Zein 1998 , Triono 1999 , Andrias 2001 , and Handoyo 2001 . The physical models used are groundwater flow physical model in between two waterbodies, groundwater flow with impervious border on the right and left side, and groundwater flow with well input with various aquifer system. All validations resulted in GWPhreatic being a valid simulator for every cases. Aside from those validation endeavours, there is also a physical development protocol which have not been validated yet with GWPhreatic, which are groundwater flow through stratified aquifer protocol by Kaeni 2015 and perched aquifer by Saragih 2015 . This study will compare both physical models to evaluate GWphreatic accuration and compare it with a trusted commercial program. The commercial program used as comparsion is GeoStudio 2016 which uses finite element algorithm method, and it provides free lisence for academic purpose, although with some limitations. This study aims to assess how accurate the program GWPhreatic and GeoStudio 2016 compared with physical model and also to assess the accuracy of GWPhreatic compared with GeoStudio 2016. Both are assessed with their respective bias value with respect to physical model. GWPhreatic bias value on stratified aquifer is 0.0049 and 0.0005 on perched aquifer. GeoStudio bias value on both aquifer, stratified and perched, are 0.0069 and 0.0042 respectively. Those values are considered very small, so it then can be concluded that GWPhreatic is reliable enough to be used to simulate groundwater flow through stratified and perched aquifer, almost as reliable as GeoStudio."