Several snack bars formulation.
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For this purpose, the names of functional foods, nutraceutical foods, pharmaceutical foods, designer foods, farmafoods, vitafoods, foodaceutical, etc. were used [1]. Other terms, often grouped separately, may be included in the same regulations: medical foods, dietary supplements, fortified foods, and botanicals.
There are definitions around the world for functional foods, but there is no official or accepted definition; the functional foods are more of a concept than a defined food product group. Therefore, a functional food can be a natural food, a food in which a component has been added or has been removed, a food in which the nature of one or more components has been altered, or any combination of these possibilities [2]. Functional foods can be those foods made up of distinct, fortified, and enriched elements that provide health benefits apart from the supply of essential nutrients (e.g., vitamins and minerals) when consumed at effective levels within a varied diet [3]. A food may be functional for all individuals or for certain population groups, which can be defined, for example, by age or genetic constitution.
Moreover, the functional food, besides its basic nutritional effect, has also beneficial effects on human body functions (improving the general physical state, decreasing the risk of disease evolution, etc.) [1, 4, 5, 6]. The main roles of functional foods in regulating processes in the body are presented in Figure 1.
The role of functional foods in regulating processes in the body.
For several years now, there has been a substantial interest in fast food and snack food production, due to changes on people’s lifestyles. Consumers have the tendency to look for easily prepared food, such as snacks, which are defined as alternatives to quick meals with or without substantial nutritional value. According to Bower and Whitten [7], numerous products are classified as “snacks,” and in this category, mini-pizzas, cakes, popcorn, cereals, and cereal-based bars can be included.
Generally, snack bars are not recognized as functional foods, mainly due to their nutrient-poor composition. In the last years, there is an interest in making new types of snack bars with functional components. Therefore, snack bars can also be included in the functional product category and to consider consumer’s acceptable and suitable ready-to-eat products.
According to a report regarding the snack food consumption in the USA [8], the snack, cereal, and nutrition bars may be classified in three main categories: health and wellness snack, organic snack bars, and energy and nutrition bars. A complex classification of functional bars is difficult to obtain, and the focus has generally been on consumption. Therefore, snack bars can be consumed as a meal part (as part of breakfast, lunch, or dinner or as a snack between meals), as a dessert (after lunch or dinner), or as a meal replacement (breakfast, lunch, or dinner).
The consumption of the snack bars is usually influenced by the age, gender, and the nutritional knowledge of the consumers. According to the International Markets Bureau market indicator [8], the consumption of the snack bars is also influenced by the following aspects: satisfying the need for sweets; saving time; using as an energy source; using for weight loss; and using for the protein, fiber, vitamin contents, etc.
The simplified flowcharts of snack bar production processes are shown in Figure 2. For the baked bar products, dry and wet raw materials are mixed together. This mixture is portioned and subjected to baking.
Flowchart of snack bar production process (I, baked bars; II, baked bars with filling; III, cold-formed snack bar).
The baking parameters (time and temperature) differ according to the specific characteristics of the finished product. For cold-formed bar production, the ingredients are also mixed together, and the resulting mixture is portioned in the desired shape without a baking step. For both types of snacks, there may also be additional operations, e.g., filling, coating with various glazes, drying bars, etc. Despite the bar type obtained, the final operation in the technological chart is packing.
The bars are often made using a base of cereals such as oats, rice, corn, or proteins (milk dairy proteins, soy, or whey) and fortified with vitamins, minerals, and other nutrient- or energy-rich ingredients [9, 10]. Snack bars are not just popular for their portability but also for the health implications associated with their consumption. Trends in health and wellness in food and beverages have increased more and more in recent years, as consumers are turning to less processed and more natural alternatives than regular products. Furthermore, nutritional snacks are suitable products that can provide energy and micronutrients to both healthy people and people from areas affected by famine of the world [10].
Cereal bars are very adaptable products made from processed cereals mixed with a variety of ingredients depending on the target population group [7]. Wheat and/or soy snack bars were designed as nutritional bars to provide the nutrients to consumers on the run [10]. Walnuts were successfully used in the manufacture of snack bars with good nutritional (significant amount of raw fibers and lipids) and sensory quality [11].
In recent years, the demand for high-protein snack bars has grown significantly by the people engaged in sports activities and dieting and as meal substitutes. These snack bars provide a healthy alternative to conventional snacks due to its high content of protein (15–35%, w/w) and other nutritionally beneficial ingredients [12]. In Table 1, different formulations for snack bars are shown.
Snack bar type | Formulation and Functionality | References |
---|---|---|
Fruit-based snack bar | Ingredients: Base: rice crisps, glucose syrup, honey, vegetable oil, quick-cook rolled oats (or natural apple dietary fibre or inulin), glycerol, whey protein concentrate, maltodextrin, pectin; Filling: pectin, sugar, citric acid, glucose syrup, honey, vegetable shortening, glycerol, and apple purée. Process: Base: the dry and wet ingredients were mixed, moulded and baked at 130°C/15 min; Filling: ingredients were heated with stirring to 84-86% soluble solids content; Assembly: the filing was placed between base parts Functionality: increased polyphenols and dietary fibre contents | [13] |
Wheat- or soy-based bar | Ingredients: water, corn syrup, glycerine, brown sugar, Arabic gum, emulsifier, puffed wheat, wheat germ, commercial coatings, shortening, soy nuggets, soy protein, soy fibre, vanilla extract Process: ingredients (water, corn syrup, glycerine, brown sugar, Arabic gum, emulsifier, shortening, vanilla extract) were mixed and blended in different proportions to obtain wheat bar; wheat bar with coating; wheat and soy bar with coating; soy bar with coating; wheat bar with glycerine. The puffed wheat, wheat germ, soy nuggets, soy protein, soy fibre, were heated at 85°C/4 min. All the ingredients were mixed together and moulded. The bars were cooled at room temperature. Some bars were coated and cooled supplementary. Functionality: high quality proteins, fibres and B-complex vitamins contents | [10] |
Cereal snack bar | Ingredients: corn starch biscuit (or marolo flour), skimmed milk powder, rice flakes, oat flakes, corn syrup Process: dry ingredients were mixed and then syrup was added. The bars were manually sealed, cooled for 8 h and then cut and packed in aluminium-coated cellophane Functionality: the marolo flour provided an increase in dietary fibre content, vitamin C, minerals, and antioxidant activity. | [14] |
Fruit and vegetable-based snack bar | Ingredients: Sweet: fruit and vegetable flour, rice flour, oat flakes, linseeds, sucrose syrup, brown sugar, cocoa powder Salty: fruit and vegetable flour, rice flour, oat flakes, linseeds, egg white, olive oil, condiments Process: Sweet: the sucrose syrup heated was mixed with dry ingredients; Salty: ingredients were mixed, moulded into bars and baked at 160°/10 min, and reposed for 6h at room temperature until cold; Both types were coated in aluminized bags, sealed Functionality: high fibre, protein and mineral contents | [15] |
Cereal snack bar | Ingredients: crushed corn starch biscuit (or jerivá flour), oat, skimmed milk powder, rice flakes, corn syrup Process: dry ingredients were mixed, adding the syrup (corn glucose and honey) and mixing. The mixture was plastic-coated and reposed for 8h, then moulded Functionality: increasing in total dietary fiber, vitamin C, mineral contents and antioxidant activity | [16] |
Vegetable based snack bar | Ingredients: bean flour, oat flakes powder, water and soy lecithin Process: the ingredients were mixed, and the dough obtained was coated and cut to a standard size. The snacks were baked using convection oven at 120°C/30 min, cooled at room temperature and packed in metallic bags Functionality: increasing in protein and dietary fiber content and the antioxidant capacity | [17] |
High-protein snack bar | Ingredients: milk protein, high fructose corn syrup and glycerol Process: the ingredients were mixed, and the mixture obtained was moulded into plastic cups. Functionality: increasing in protein content | [12] |
Several snack bars formulation.
The tendency to eat more nutritious foods instead of sweet products has led to the development of different snack bar types. Since cereal consumption extends beyond breakfast at any time of the day, these products have become an excellent vehicle for delivering ingredients to functional foods on the market. Cereals have an increasingly important role in modern lifestyle due to the convenient forms they can use such as ready-to-eat food products, snack bars, and energy bars [14].
People who are interested in getting healthier foods and maintaining good body fitness have changed their eating habits, which have promoted growth in the cereal bar market of 20% per year [18]. Therefore, snack bars can be considered as a reliable source of high-quality proteins, fibers, vitamins, and minerals. The nutritive composition of selected snack bars is presented in Table 2.
From the data presented in Table 2, the chemical composition of the snack bars varies greatly depending on their destination. The variation of snack bar composition gives the consumers the possibility to select the proper bar to purchase. Therefore, corn flake crust with fruit breakfast bar is a good source of vitamin A, thiamine, riboflavin, niacin, and vitamin B6, while formulated bar (power bar with chocolate) is a good source of protein, vitamin E (alpha-tocopherol), calcium (covers 50% of the recommended daily dose), iron, magnesium, phosphorus, zinc, and copper. Moreover, the power bar is a very good source of vitamin C (covers 154% of the recommended daily dose), thiamine (covers 537% of the recommended daily value), riboflavin, niacin, vitamin B6, folate, vitamin B12, pantothenic acid, and manganese [19].
Nutrients | Unit/100g | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|
Proteins | g | 9.8 | 4.4 | 14.2 | 18.5 | 25 | 21.9 | 17.5 | 12.46 |
Lipids | g | 17.6 | 7.5 | 3.1 | 13.2 | 10 | 10.8 | 17.5 | 14.82 |
Carbohydrates | g | 66.7 | 72.9 | 69.6 | 55.1 | 53 | 50.3 | 52.5 | 62.01 |
Total fibers | g | 3.1 | 2.1 | 5.7 | 2.8 | 10 | 6.7 | 5 | 20.84 |
Minerals | |||||||||
Calcium, Ca | mg | 60 | 41 | 504 | 953 | 625 | 754 | 15 | ~ |
Iron, Fe | mg | 3.18 | 4.9 | 11.7 | 14.7 | 10.1 | 16.3 | 5 | ~ |
Magnesium, Mg | mg | 101 | 27 | 248 | 255 | 175 | 317 | 0 | ~ |
Phosphorus, P | mg | 277 | 103 | 614 | 455 | 375 | 455 | 2.5 | ~ |
Potassium, K | mg | 326 | 197 | 362 | ~ | 300 | 351 | ~ | ~ |
Sodium, Na | mg | 251 | 167 | 308 | 418 | 225 | 383 | 125 | ~ |
Zinc, Zn | mg | 1.6 | 4.1 | 10 | 9.6 | 6.6 | 11.9 | 0 | ~ |
Vitamins | |||||||||
Vitamin C, total ascorbic acid | mg | 1 | 0 | 92.3 | 144 | 75 | 490 | 0 | ~ |
Thiamine | mg | 0.28 | 1.0 | 8.06 | 2.7 | 1.9 | 29 | 0 | ~ |
Riboflavin | mg | 0.11 | 1.1 | 1.92 | 3.1 | 2.1 | 3.3 | 0 | ~ |
Niacin | mg | 1.75 | 13.5 | 32.6 | 36.4 | 25 | 45.2 | 0 | ~ |
Vitamin B-6 | mg | 0.35 | 1.4 | 2.2 | 3.6 | 2.5 | 4.5 | 0 | ~ |
Folate, DFE | μg | 81 | 108 | 1046 | 727 | 500 | 816 | 0 | ~ |
Vitamin B-12 | μg | 0 | 0 | 5.6 | 10.9 | 7.5 | 12.2 | 0 | ~ |
Vitamin E (alpha-tocopherol) | mg | 0.82 | 0.8 | 8.28 | 24.6 | 125 | 27.5 | 0 | ~ |
Moisture | g | 4.1 | 14.5 | 9.8 | 8.0 | 9.0 | 14.0 | ~ | 6.8 |
Ash | g | 1.8 | 0.8 | 3.3 | 3.1 | 3.0 | 3.0 | ~ | 3.89 |
Energy value | Kcal | 464 | 377 | 363 | 406 | 402 | 386 | 450 | 347.9 |
Chemical composition of selected snack bars made with cereal, fruits, nuts and/or chocolate [10, 17, 19].
1, breakfast bars, oats, sugar, raisins, coconut (include granola bar); 2, breakfast bar, corn flake crust with fruit; 3, formulated bar, power bar, chocolate; 4, formulated bar, marathon multi grain crunch bar; 5, formulated bar; protein performance bar, caramel nut rush; 6, formulated bar, marathon energy bar; 7, snack bar with soy; 8, snack bars with beans and oat flour; ~ n.a., not analyzed.
Also, formulated bar, marathon protein performance bar and caramel nut rush, is a good source of protein (covers 50% of the recommended daily dose), calcium, iron, magnesium, zinc, copper, and manganese and a very good source of vitamin C, vitamin E (alpha-tocopherol), thiamine, riboflavin, niacin, vitamin B6, folate, vitamin B12, and pantothenic acid [19], while snack bar with soy can be considered as a good source of high-quality proteins, fibers, and B-complex vitamins [10].
Researchers reported that snack bars with a high ratio of protein/carbohydrate can improve post meal and diurnal glucose profiles in patients with type 2 diabetes and insulin resistance [20]. In addition, proteins can have a very positive effect on the human body, helping to control blood pressure, and the effect has been dependent on the type of protein [18]. Champ et al. [21] reported that the demand to increase dietary fiber content in the daily diet is justified by the positive role of fibers in health and disease prevention, especially in digestive health, energy balance, cancer, and heart and diabetes problems. According to the data presented in Table 2, some formulated snack bars contain higher amounts of minerals and vitamins. Micronutrients are necessary compounds for a proper physiological state of the body that can be administered orally in the diet, and it is necessary to maintain an adequate balance [21, 22]. Minerals must be supplied from foods because they cannot be synthesized, and given the global deficiencies in different minerals, it is worthy to offer convenient food with sufficient amounts of minerals [23].
From the compounds present in snack food, only a few, through their specific action, are essential to life and are usually known as biologically active compounds. Proteins, polyunsaturated fatty acids, vitamins, and minerals as well as food fibers or probiotics may be included in this category.
The importance of dietary protein, as a bioactive factor, is determined deficit terms, by decreasing of metabolic capacity, reduction of energy metabolism, inhibition of biosynthesis processes, diminution of immunity, and the body resistance to external agent’s actions. Proteins are necessary to support growth, tissue repair, and protection [24]. Usually, the protein necessary is influenced by the gender, age, activity level, body health, or physiological states [25, 26]. The conventional protein sources are plants and animals, e.g., meat, milk, whey egg, fish, soy, etc. [26]. In general, protein-based snack bars contain the following proportion of ingredients: soy or dairy proteins (20–40%), carbohydrates as sugar syrups (10–50%), and fats (10–15%) [27, 28]. Protein bars in addition to basic ingredients may contain other components such as flavors and stabilizers [29].
In addition to their role as a source of high calories, as a biomembrane construction material, or as a vehicle of fat-soluble vitamins, lipids are important by their intake in polyunsaturated fatty acids, phosphatide, and substances physiologically active, such as prostaglandin, prostacyclin, etc. [30]. The omega-3 (n-3) and omega-6 (n-6) polyunsaturated fatty acids are essential for human health and are obtained exclusively from the nutrition [31]. In recent years, emphasis has been placed on obtaining functional foods supplemented with n − 3 fatty acids. At present, there is a wide range of omega-3-enriched food products such as bakery products, dairy products, juices and soft drinks, meat products, etc. [32]. Usually, the natural sources of n-3 fatty acids are seed and fish oils [31, 33]. Snack bars with a high content of essential fatty acids are generally those that contain crop seeds and vegetable oils (canola, soybeans, corn, and sunflower oils); these are the major sources of linoleic acid (LA, C18: 2) but with a low proportion of α-linolenic acid (ALA, C18: 3) [34]. ALA sources are chia and flaxseed [34, 35]. The flaxseeds have a high antioxidant potential and omega-3 α-linolenic acid content [36, 37]. The addition of milled flaxseed in snack products, such as bars, is explained by their preventive and functional properties, such as vascular function improvement [38], anti-arrhythmic, anti-atherogenic, and anti-inflammatory functions [38, 39, 40].
Mineral elements have a biological key role because they participate in all the vital processes of the body. Mineral substances can have different roles in vitamins and enzyme functionality, having either an activating role (calcium, magnesium, etc.) or an inhibitor (copper), and can be a part of enzyme structure as specific (sulfur, iron, copper, zinc, etc.) or unspecific (magnesium, zinc, cobalt, etc.) elements [41]. The commercial snack bars usually contain the following minerals in different intakes (Table 2): calcium, iron, magnesium, phosphorus, potassium, sodium, and zinc.
Vitamins are biologically active compounds of organic nature, with variable and complex structure. Vitamins are needed in very small quantities for the normal development of many metabolic processes, including assimilation and the use of nutrients brought by food, growth, and tissue restoration.
Usually, snack bars commercialized on the market are rich in vitamins such as vitamin B1 (thiamine), vitamin B2 (riboflavin), vitamin B5 (pantothenic acid), vitamin B3 (niacin), vitamin B6 (pyridoxine, pyridoxal, pyridoxamine), folic acid (pteroylglutamic acid, vitamin B9), vitamin B12, vitamin C, vitamin A, and vitamin E (alpha-tocopherol).
Vitamin B1 is found in the outer layers and germ of cereals, nuts, whole grains, brewer’s yeast, etc. Vitamin B1 exhibits several functions within the body, such as essential for maintaining appetite and normal digestion and necessary for growth, fertility, and lactation [42], as well as for the normal functioning of the nervous tissue [43]. According to the nutrition data [19], the bars with the highest content in thiamine are formulated bar, power bar, chocolate (8 mg/100 g), and formulated bar, marathon types: honey nut oat bar; energy bar, all flavors; chewy chocolate peanut bar; and multigrain crunch bar (~3 mg/100 g), with high daily values for adults between 197 and 537%.
Vitamin B2 is widely found in food products. Sources rich in B2 are beer and bakery yeast, meat, eggs, fish, and dairy products. In the body, vitamin B2 fulfills certain metabolic functions, as phosphorus esters are part of the flavin enzyme structure involved in biological oxidation–reduction reactions [44]and together with pyridoxine is involved in the conversion of tryptophan to nicotinic acid.
Vitamin B5, in the form of coenzyme A, is vital in the metabolism of lipids, carbohydrates, and nitrogen compounds [44]. Sources rich in pantothenic acid that are used in snack bar composition are wheat germ, mushrooms, broccoli, cauliflower, oats, dried peas, soybean, etc.
Vitamin B6 is found in large quantities in beer and bakery yeast and raw cereals. Fruits and vegetables have a lower content of vitamin B6. Pyridoxal phosphate and pyridoxamine phosphate are coenzymes involved in transamination, decarboxylation, deamination, etc. [45].
In the body, folic acid as a coenzyme is involved in accepting and transferring one carbon, from a metabolite to another [46], interfering with the synthesis of purines and pyrimidines necessary for nucleic acid formation [47]. Folic acid is mainly found in green leaves, whole grains, soybeans, wheat germ, soybeans, oranges, avocado, etc. [47, 48].
Vitamins E performs several functions in the body such as Krebs cycle regulator, a regulator of nucleic acid metabolism with implications for cell maturation and differentiation, a regulator of porphyrin and heme biosynthesis, a regulator of protein and amino acid metabolism, etc. The main nutritional function is a biological antioxidant for oxidizable vitamins and lipids [44]. Tocopherols are found in almost all foods, the richest sources usually used for bar production being vegetable oils (corn germ oil, wheat germ oil), cereal products, and eggs.
Vitamin C is acting as biochemical antioxidant and as enzyme cosubstrate [44], interfering with numerous processes, including formation of stromal proteins (collagen, elastin), maintaining capillary integrity, etc. Sources rich in vitamin C are rosehips, white sea buckthorn, citrus fruits, tomatoes and tomato juice, cabbage, parsley, nettle, spinach, cauliflower, red and green pepper, red cabbage, eggplant, broccoli, strawberries, etc. [45, 49]. In a study conducted by Sung et al. [50], a cereal bar with welsh onion extracts was obtained, rich in vitamins C, B2, B3, and B9 and with high-protein content. The study also demonstrated the nutraceutical potential of the bars for the obesity and metabolic disorder’s control.
USDA Food Composition Databases [51] provides the selection of snack bars according to deficiency in one or more vitamins or any nutrient that must be reintroduced into the diet. As an example, a selection of the snack bar products according to the content of thiamine, riboflavin, and vitamin C is shown in Table 3.
Description | Thiamine (mg/100 g) | Riboflavin (mg/100 g) | Vitamin C (mg/100 g) |
---|---|---|---|
Formulated bar, power bar, chocolate | 8.06 | 1.92 | 92.3 |
Formulated bar, snickers marathon | |||
Honey nut oat bar | 3.394 | 3.846 | 176.4 |
Energy bar, all flavors | 2.949 | 3.343 | 489.9 |
Chewy chocolate peanut bar | 2.727 | 3.091 | 144.4 |
Multigrain crunch bar | 2.727 | 3.091 | 144.3 |
Protein performance bar, caramel nut rush | 1.875 | 2.125 | 75 |
Formulated bar, meal bar, milk chocolate peanut | 1.7 | 1.9 | 96.6 |
Snack, balance, original bar | 1.628 | 1.708 | 11.2 |
Snacks, crisped rice bar, almond | 1.323 | 1.499 | 11.5 |
Snacks, nutri-grain cereal bars, fruit | 1.153 | 2.067 | 7.8 |
Breakfast bar, corn flake crust with fruit | 1 | 1.1 | 0 |
Snacks, granola bar, with yogurt coating | 0.905 | 0.542 | 0 |
Formulated bar, mixed flavors | 0.75 | 1.7 | 120 |
Classification of snack bars according to the content of thiamine, riboflavin and vitamin C [51].
In the food industry, fiber plays a significant role in preventing obesity, diabetes, and cardiovascular disease [52]. Fiber is incorporated into different foods for both nutritional benefits and functional properties, as well as low-cost and noncaloric partial replacements for fat and/or sugar flour [52, 53]. Cereals contain quantities of insoluble fiber, except for oats, whereas fruit and, in particular, citrus fruit and apples contain substantial amounts of soluble fiber. Lentils are also used in snack bar development, especially for their dietary fiber and the essential amino acid contents and for the folic acid and iron addition [54].
Lately, emphasis has been placed on the use of resistant starch (RS) in products as a potential functional ingredient. RS may increase the dietary fiber content [55] and promote the probiotic bacteria activity [56]. Aigster et al. [57] studied the addition of RS in granola and cereal bars, to determine the RS health significance and also the consumer acceptability. The study concluded the increasing of dietary fiber intake by the addition of RS.
In recent years, consumers are constantly changing their eating habits, wanting to improve their diet with nutritious and safe food products that are easy to consume but, at the same time, improving health and well-being. Therefore, food industry specialists have continuously changed formulations and ingredients, and thus, new technologies to produce foods with increasing nutritional value and safety that fully meet consumers’ requirements. Due to their versatility, snack bars can be used by wide categories of individuals: athletes, people dieting or with nutritional problems, or irregular meals. Snack bars allow the consumers to get necessary energy and the nutritional benefits (high-quality proteins, polyunsaturated fatty acids, minerals, vitamins, and fibers) in one convenient and easy to store package.
The authors declare no conflict of interest.
During an event of earthquake stability of dams is of paramount importance as their failure can cause immense property and environmental damages. When dam-reservoir-foundation system is subjected to the dynamic loading it causes a coupled phenomenon; ground motion and deformations in the dam generate hydrodynamic pressure in the reservoir, which, in turn, can intensify the dynamic response of the dam. Moreover, spatial-temporal variation of stresses in the dam-body depends on the dynamic interactions between the dam, reservoir, and foundation. Therefore, it becomes necessary to use numerical techniques for the safety assessment of a given dam-design against a particular ground motion.
\nDynamic finite element method is the most extended approach for computing the seismic response of the dam-reservoir system to the earthquake loading [1]. In this approach finite elements are used for discretization of space domain, and basis functions are locally supported on the spatial domain of these elements and remain independent of time. Furthermore, nodal values of primary unknowns depend only on time. Accordingly, this arrangement yields a system of ordinary differential equations (ODEs) in time which is then solved by employing time-marching schemes based on the finite difference method (FDM), such as Newmark-
In dynamic finite element method (FEM), it is desirable to adopt large time-steps to decrease the computation time while solving a transient problem. Therefore, it is imperative that the time-marching scheme remains unconditionally stable and higher order accurate [1]. In addition, it should filter out the high frequency components from the response of structure. To achieve these goals, Hughes and Hulbert presented space-time finite element method (ST/FEM) for solving the elastodynamics problem [2]. In this method, displacements
However, for elastodynamics problem, ST/FEM, yields a larger system of linear equations due to due to the time-discontinuous interpolation of displacement and velocity fields. Several efforts have been made in the past to overcome this issue; both explicit [8] and implicit [9] predictor-multi-corrector iteration schemes have been proposed to solve linear and nonlinear dynamics problems. Recently, to reduce the number of unknowns in ST/FEM, a different approach is taken in which only velocity is included in primary unknowns while displacement and stresses are computed from the velocity in a post-processing step [4, 10]. To this end, the objective of the present chapter is to introduce this method (henceforth, ST/FEM) in a pedagogical manner. The rest of the chapter is organized as follows. Sections 2 and 3 deal with the fundamentals of time-discontinuous Galerkin method. Section 4 describes the dam-reservoir-soil interaction problem, and Section 5 discusses the application of ST/FEM for this problem. Lastly, Section 6 demonstrates the numerical performance of proposed method and in the last section concluding remarks are included.
\nConsider a mass-spring-dashpot system as depicted in Figure 1. The governing equation of motion is described by the following second order initial value problem in time.
\nSchematic diagram of the mass-spring-dashpot system.
where
In what follows, this second order ODE will be utilized to discuss the fundamental concepts behind time-discontinuous Galerkin methods (henceforth, tDGM).
\nIn two-field tDGM (henceforth, uv-tDGM), both displacement (
Schematic diagram of time discontinuous approximation: (a) piecewise linear interpolation, and (b) piecewise quadratic interpolation.
where
\nare the discontinuous values of
The weak-form of the uv-tDGM can be stated as: find
where
Eq. (10) denotes that, in uv-tDGM, displacement-velocity compatibility relationship is satisfied in weak form.
\nTo decrease the number of unknowns in comparison to those involved in uv-tDGM, displacement-velocity compatibility condition (cf. Eq. 6) can be explicitly satisfied and velocity can be selected as primary unknown. Henceforth, this strategy will be termed as v-tDGM. In v-tDGM,
The weak form of the v-tDGM reads: Find
Note that Eqs. (9) and (11) are identical, however, in former,
Let us now focus on the discretization of weak-form (cf. Eq. (11)) by using the locally defined piecewise linear test and trial functions,
\nwhere
\nAccordingly, Eq. 11 transforms into following matrix-vector form.
\nwhere
In this section, numerical analysis of the tDGM schemes, (viz. uv-tDGM and v-tDGM) for the second order ODE will be performed. To assess the stability characteristics and temporal accuracy of these schemes, classical finite difference techniques will be used ([11], Chapter 9). In this context, it is sufficient to consider the following homogeneous and undamped form of Eq. (1):
\nIn this section it will be shown that v-tDGM is a true energy-decaying scheme. Consider Eq. (17) which represents the governing equation of a spring-mass system. The total energy (sum of kinetic and potential energy) of the system remains constant because damping and external forces are absent in the system.
\nConsider the time domain
Accordingly, it can be shown that
\nor
\nThis shows that v-tDGM is an energy decaying time integration algorithm, in which the total energy during any time step,
To assess the energy dissipation characteristics of v-tDGM, Eq. (17) is solved with
Energy decay characteristics of v-tDGM; (a) temporal variation of normalized total energy and (b) phase diagram obtained with different time-step sizes.
In this section, to study the stability characteristics of v-tDGM, Eq. (17) is considered. The matrix-vector form corresponding to this problem is given by
\nwhere
where
To investigate the stability of v-tDGM one should look into the eigenvalues of
Figure 4 plots the frequency responses of
Frequency response of spectral radius ρ for v-tDGM.
In [4], it is shown that
where
Subsequently, by expanding
A direct consequence of the convergence is that the solution of Eq. (17) can be given by following expression [1]:
\nwith
\nwhere
Further, to investigate the accuracy of v-tDGM, algorithmic damping ratio, which is a measure of amplitude decay, and relative frequency error
Accuracy of v-tDGM: (a) algorithmic damping ratio, and (b) relative frequency error in low frequency regime (after [4]).
A dam-reservoir-soil (DRS) system which is subjected to the spatially uniform horizontal (
Schematic diagram of dam-reservoir-soil (DRS) system subjected to seismic ground motion.
Further, hydrodynamic pressure distribution in the reservoir is modeled by the pressure wave equation,
\nwith following initial and boundary conditions.
\nIn Eq. (26),
Let us now consider the initial-boundary value problem of the solid domain which is described by,
\nFurthermore, following time dependent boundary conditions will be considered in Eq. (33):
\nIn addition, solid domain is considered to be an isotropic, homogeneous, linear elastic material with
\nIn Eqs. (31)–(34),
in which,
Recently, ST/FEM is employed to solve dam-reservoir-soil interaction problem [10] in which the resultant matrix-vector form is given by
\nwhere
Further, In Eqs. (41) and (42),
\nwhere
\ndenote mass matrix, diffusion matrix, and viscous boundary at the upstream truncated boundary, respectively. The fluid-solid coupling matrix
and right hand side spatial nodal vectors in Eqs. (41) and (42) becomes,
\nwhere
in which
the solid-fluid coupling matrix,
\nand right hand side spatial nodal vectors is given by,
\nwhere
Further, the force vector
Lastly, nodal values of displacement and pressure field at time
In this section, ST/FEM with block iterative algorithm has been employed to study the response of the concrete gravity dam to the horizontal earthquake motion (see [10]). In the numerical modeling two cases are considered; (i) dam-reservoir (DR) system, in which foundation is considered to be rigid, and (ii) dam-reservoir-soil (DRS) system, in which the foundation is an elastic deformable body.
\nFigure 7 depicts the physical dimensions of the dam-reservoir system. Length of the reservoir in upstream direction is
Physical dimensions of dam-reservoir system (after [10]).
Figure 8 represents the accelerogram (horizontal component) recorded at a control point on the free surface; the maximum and minimum values of acceleration are
Time history of horizontal component of ground motion recorded at free-surface.
Acceleration response at the crest of dam in DR and DRS system; time history of (a) horizontal component and (b) vertical component of acceleration, and Fourier spectrum of (c) horizontal component and (d) vertical component of acceleration (after [10]).
Interestingly, in both cases, it is observed that the critical location for pressure is at the base of the dam. Figure 10 presents the evolution of
Temporal response of (a) normalized hydrodynamic pressure, (b) principal tensile stress and (c) principal compressive stress at the base of dam in DR and DRS system (after [10]).
(a) and (c): Hydrodynamic pressure field in the reservoir, and (b) and (d): magnified deformed configuration of dam at times t = 18:02 s and t = 18:08 s (after [10]).
In this chapter, novel concepts of time-discontinuous Galerkin (tDGM) method is presented. A method called v-tDGM is derived to solve second order ODEs in time. In this method velocity is the primary unknown and it remain discontinuous at discrete times. Thereby, the time-continuity of velocity is satisfied in a weak sense. However, displacement is obtained by time-integration of the velocity in a post-processing step by virtue of which it is continuous in time. It is demonstrated that the present method is unconditionally stable and third-order accurate in time for linear interpolation of velocity in time. Therefore, it can be stated that the numerical characteristics of the v-ST/FEM scheme, therefore, make it highly suitable for computing the response of bodies subjected to dynamic loading conditions, such as fast-moving loads, impulsive loading, and long-duration seismic loading, among others.
\nSubsequently, ST/FEM is used to compute the response of a dam-reservoir-soil (DRS) system to the earthquake loading while considering all types of dynamic interactions. An auxiliary variable
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